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This article is part of the series Proceedings of International Conference on Applied Analysis and Mathematical Modeling 2013.

Open Access Research

Approximation on the hexagonal grid of the Dirichlet problem for Laplace’s equation

Adiguzel A Dosiyev* and Emine Celiker

Author Affiliations

Department of Mathematics, Eastern Mediterranean University, Gazimagusa, K.K.T.C., via Mersin, 10, Turkey

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Boundary Value Problems 2014, 2014:73  doi:10.1186/1687-2770-2014-73

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/73


Received:28 November 2013
Accepted:9 March 2014
Published:28 March 2014

© 2014 Dosiyev and Celiker; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

Abstract

The fourth order matching operator on the hexagonal grid is constructed. Its application to the interpolation problem of the numerical solution obtained by hexagonal grid approximation of Laplace’s equation on a rectangular domain is investigated. Furthermore, the constructed matching operator is applied to justify a hexagonal version of the combined Block-Grid method for the Dirichlet problem with corner singularity. Numerical examples are illustrated to support the analysis made.

MSC: 35A35, 35A40, 35C15, 65N06, 65N15, 65N22, 65N99.

Keywords:
Laplace’s equation; Dirichlet boundary value problem; hexagonal grids; matching operator; interpolation for harmonic functions; singularity; Block-Grid method

1 Introduction

When solving PDEs, many approximate methods, such as overlapping versions of domain decomposition, composite grids and different types of combined methods, use the matching operator to connect the subsystems within them. Hence, the approximation of the solutions relies heavily on the order of accuracy of the matching operator, as well as on the order of accuracy of the subsystems. In [1] and [2], the second order matching operator is used to construct and justify the second order composite grid method for solving Laplace’s boundary value problems. In [3], the fourth order matching operator is constructed and used for the fourth order composite grids and in [4] and [5] it is used for the fourth order Block-Grid method. In [6-9], the sixth order matching operator is constructed for the Block-Grid method and it is used for the sixth order composite grids in [10].

In all of the above mentioned papers, the fourth and sixth order matching operators were constructed on the basis of the 9-point finite difference solution of Laplace’s equation on square grids. In this paper, the matching operator is constructed for the solution of the Dirichlet problem on a hexagonal grid. In order to approximate the given differential equations at each regular node <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M1">View MathML</a> on a hexagonal grid, the six equidistant nodes surrounding <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M1">View MathML</a> are used, and the truncation error obtained is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M3">View MathML</a>. Thus, we obtain the same order of accuracy when using the 7-point scheme on the hexagonal grid, as we do when using the 9-point scheme on the rectangular grid (see [11]). This has many computational advantages such as (i) the matrix of the system will contain seven diagonals rather than nine and will lead to less use of memory space, (ii) the calculations will require less computational effort and (iii) the algorithm will be easier to implement. Hexagonal grids are favored in many applied problems in dynamical meteorology and dynamical oceanography as well (see [12-14]), due to the benefits a hexagonal grid provides, compared to a rectangular grid.

Even though using a hexagonal grid has the above mentioned advantages, it has not been used before in methods such as composite grids, domain decomposition, and combined methods, as the fourth order matching operator for connecting the subsystems together was not constructed. In this paper, in Section 2, the approximate solution in a hexagonal grid on a rectangular domain is analyzed. In Section 3 a fourth order matching operator is constructed and its application to find the fourth order accurate approximate solution on the closed domain is considered. Section 4 contains the justification of using a hexagonal grid for the solution of Laplace’s equation on a staircase polygon, with the use of the Block-Grid method. This method requires the application of the matching operator constructed in Section 3 and gives an overall fourth order accuracy. Numerical examples are illustrated in Section 5 to support the analysis made.

2 Approximation in the hexagonal grid of the Dirichlet problem on a rectangle

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M4">View MathML</a> be a rectangle, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M6">View MathML</a>, be its sides, including the ends, enumerated counterclockwise starting from left (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M8">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M9">View MathML</a> be the boundary of Π, and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M10">View MathML</a> be the jth vertex. We consider the boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M11">View MathML</a>

(2.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M12">View MathML</a>

(2.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M13">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M14">View MathML</a> is a given function of arclength s taken along γ, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M15">View MathML</a>

(2.3)

At the vertices <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M16">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M17">View MathML</a> is the beginning of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M5">View MathML</a>), the conjugation conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M19">View MathML</a>

(2.4)

are satisfied.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M20">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M21">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M22">View MathML</a> integers. We assign <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M23">View MathML</a> a hexagonal grid on Π, with step size h, defined as the set of nodes

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M24">View MathML</a>

(2.5)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M25">View MathML</a> be the set of nodes on the interior of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M5">View MathML</a>, and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M27">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M28">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M29">View MathML</a>. Also let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M30">View MathML</a> denote the set of nodes whose distance from the boundary γ of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M31">View MathML</a> is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M32">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M33">View MathML</a>.

We consider the system of finite difference equations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M34">View MathML</a>

(2.6)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M35">View MathML</a>

(2.7)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M36">View MathML</a>

(2.8)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M37">View MathML</a>

(2.9)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M38">View MathML</a>

(2.10)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M39">View MathML</a>

(2.11)

From formulae (2.9) and (2.10) it follows that the coefficients of the expressions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M40">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M41">View MathML</a> are non-negative, and their sums do not exceed one. Hence, on the basis of the maximum principle, it follows that the solution of system (2.6)-(2.8) exists and it is unique (see [11]).

Everywhere below we will denote constants which are independent of h and of the cofactors on their right by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M42">View MathML</a> , generally using the same notation for different constants for simplicity.

Lemma 2.1Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M43">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M44">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M45">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M46">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M47">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M48">View MathML</a>are arbitrary grid functions. If the conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M49">View MathML</a>

are satisfied, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M50">View MathML</a>

Proof The proof of this lemma is similar to the proof of the comparison theorem (see Ch. 4 in [11]). □

Theorem 2.2Letube the solution of problem (2.1), (2.2) and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M51">View MathML</a>be the solution of system (2.6)-(2.8), then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M52">View MathML</a>

(2.12)

Proof Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M53">View MathML</a>

where u is the trace of the solution of problem (2.1), (2.2) on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M54">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M51">View MathML</a> is the solution of system (2.6)-(2.8). Then, the error function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M56">View MathML</a> satisfies the following system:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M57">View MathML</a>

(2.13)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M58">View MathML</a>

(2.14)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M59">View MathML</a>

(2.15)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M60">View MathML</a>

(2.16)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M61">View MathML</a>

(2.17)

are the truncation errors of equations (2.6) and (2.7), respectively.

On the basis of conditions (2.3) and (2.4), from Theorem 3.1 in [15] it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M62">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M63">View MathML</a>. Then, by Taylor’s formula, we obtain (see [16])

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M64">View MathML</a>

(2.18)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M65">View MathML</a>

We represent the solution of (2.13)-(2.15) as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M66">View MathML</a>

(2.19)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M67">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M68">View MathML</a>

By taking the function as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M69">View MathML</a> in Lemma 2.1, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M70">View MathML</a>

(2.20)

Using Taylor’s formula about each of the points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M71">View MathML</a> and from (2.17), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M72">View MathML</a>

On the basis of the maximum principle, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M73">View MathML</a>

(2.21)

From (2.19), (2.20), and (2.21) it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M74">View MathML</a>

(2.22)

 □

Remark 2.3 Estimation (2.12) remains true when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M75">View MathML</a> in system (2.6)-(2.8) is replaced by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M76">View MathML</a>

3 Construction of the fourth order matching operator

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M77">View MathML</a> be a complex variable, and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M78">View MathML</a> be a unit circle. For a harmonic function u on Ω with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M79">View MathML</a>, by Taylor’s formula, any point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M80">View MathML</a> can be represented as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M81">View MathML</a>

(3.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M82">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M83">View MathML</a>

By analogy with the idea used in [8], we construct the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M84">View MathML</a> from the condition that the expression

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M85">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M86">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M87">View MathML</a> is a node of the hexagonal grid <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M23">View MathML</a>, gives the exact value of any harmonic polynomial

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M89">View MathML</a>

at each point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M90">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M91">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M92">View MathML</a> denote the set of points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M90">View MathML</a> such that all the nodes <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M87">View MathML</a> to determine the expression <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M95">View MathML</a> belong to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M54">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M97">View MathML</a> contain the points P, where some of the nodes <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M98">View MathML</a> emerge through the side <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M6">View MathML</a>. We construct the fourth order matching operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M101">View MathML</a> by considering the cases when the point P belongs to one of the sets <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M92">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M97">View MathML</a>.

Case 1. The point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M104">View MathML</a> lies on the line connecting two neighboring grid nodes (a grid line).

We place the origin of the rectangular system of coordinates on the node <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M1">View MathML</a> and direct the positive axis of x along the grid line, so that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M106">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M107">View MathML</a>, and take the nodes.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M108">View MathML</a>

First, we find the coefficients <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M109">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M110">View MathML</a>, such that the representation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M111">View MathML</a>

(3.2)

is true for the harmonic polynomials <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M112">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M113">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M114">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M86">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M116">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M117">View MathML</a>. This gives the system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M118">View MathML</a>

(3.3)

By solving system (3.3) and rearranging (3.2) for u, we obtain the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M119">View MathML</a>

(3.4)

We now take into consideration the nodes <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M120">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M121">View MathML</a> which are symmetric to the points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M122">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M123">View MathML</a>, respectively, with respect to the x-axis. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M124">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M125">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M126">View MathML</a>, and odd with respect to y, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M127">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M116">View MathML</a>, is even with respect to y, from (3.4) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M129">View MathML</a>

Hence, we obtain the matching operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M101">View MathML</a>, for which the expression

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M131">View MathML</a>

(3.5)

gives the exact value of the harmonic polynomial <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M132">View MathML</a> at the point P, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M133">View MathML</a>

It is easy to check that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M134">View MathML</a>

(3.6)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M135">View MathML</a>

(3.7)

Remark 3.1 When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M136">View MathML</a>, the node <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M137">View MathML</a>, which is closest to P, is taken as the origin.

Case 2. The point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M104">View MathML</a> lies inside a grid cell of the hexagonal grid.

Again, we place the origin of the rectangular system of coordinates at the node <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M1">View MathML</a> and direct the positive axis of x along the grid line, so that P has the coordinates <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M140">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M141">View MathML</a>. We form an artificial grid by taking the following points:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M142">View MathML</a>

Each of the nodes <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M143">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M144">View MathML</a>, of the artificial grid falls on a grid line, and for the approximation of P the expression

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M145">View MathML</a>

is used. As <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M143">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M144">View MathML</a>, all lie on grid lines, each of these points needs to be approximated using the matching operator as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M148">View MathML</a>

From the distribution of the nodes it becomes obvious that only 17 nodes are needed for this approximation (see Figure 2).

Hence, we form the matching operator as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M149">View MathML</a>

(3.8)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M150">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M151">View MathML</a>, are defined by the coefficients obtained earlier and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M152">View MathML</a>

(3.9)

For the approximation, it is also important to examine the structure of the hexagonal grid. There are two types of triangles in each hexagon, Type A and Type B, as shown in Figure 1.

thumbnailFigure 1. Shapes of triangles in a hexagon.

We consider triangles of Type A with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M141">View MathML</a>. The nodes used in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M95">View MathML</a> are shown in Figure 2.

thumbnailFigure 2. Type A triangle with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M155">View MathML</a>.

In the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M136">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M157">View MathML</a>, the 17 nodes used have the same distribution as the reflection of the nodes in Figure 2 about the line <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M158">View MathML</a>. In the cases <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M107">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M160">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M161">View MathML</a>, the nodes for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M101">View MathML</a> are defined analogously.

In the case when P falls into a triangle of Type B, we rotate the artificial grids formed for Type A with an angle of 180, for all four cases of δ and κ specified earlier.

Case 3. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M163">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M164">View MathML</a> on the side <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M6">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M167">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M63">View MathML</a>.

We position the origin of the rectangular system of coordinates on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M5">View MathML</a> so that the point P lies on the positive yaxis, and the xaxis is in the direction of the vertex <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M170">View MathML</a> along <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M5">View MathML</a>. It is obvious that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M172">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M126">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M117">View MathML</a>. Hence, when the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M167">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M63">View MathML</a>, is represented using Taylor’s formula about the point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M158">View MathML</a> in the neighborhood <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M178">View MathML</a> of the origin, we define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M179">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M116">View MathML</a>, of (3.1) as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M181">View MathML</a>

We let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M182">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M183">View MathML</a>, and keeping in mind that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M184">View MathML</a> is odd extendable, we complete the definition with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M185">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M186">View MathML</a>. Clearly, in the given neighborhood, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M187">View MathML</a> is equal to the harmonic polynomial <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M188">View MathML</a>, with an accuracy of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M189">View MathML</a>. To form an expression for the matching operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M190">View MathML</a>, we use

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M191">View MathML</a>

(3.10)

or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M192">View MathML</a>

(3.11)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M193">View MathML</a>

(3.12)

Hence using (3.10) or (3.11), with the addition of the term

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M194">View MathML</a>

we have the following representation for the solution u of problem (2.1), (2.2) at any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M163">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M196">View MathML</a>

(3.13)

Remark 3.2 We obtain the representation (3.13), with a less number of grid nodes <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M197">View MathML</a> in (3.10) or (3.11) for the points on the boundary γ of Π.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M198">View MathML</a>. We express the matching operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M101">View MathML</a> as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M200">View MathML</a>

(3.14)

Theorem 3.3Let the boundary functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M14">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M6">View MathML</a>, in problem (2.1), (2.2) satisfy the conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M203">View MathML</a>

(3.15)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M204">View MathML</a>

(3.16)

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M205">View MathML</a>

(3.17)

whereuis the exact solution of problem (2.1), (2.2).

Proof According to Theorem 3.1 in [15], from conditions (3.15) and (3.16) it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M206">View MathML</a>. Then on the basis of (3.1), (3.5), (3.8), (3.13), and Remark 3.2, we obtain inequality (3.17). □

We define the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M207">View MathML</a> as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M208">View MathML</a>

(3.18)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M51">View MathML</a> is the solution of the finite difference problem (2.6)-(2.8).

Theorem 3.4Let conditions (2.3) and (2.4) be satisfied. Then the function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M210">View MathML</a>is continuous on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M31">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M212">View MathML</a>

(3.19)

whereuis the solution of problem (2.1), (2.2).

Proof From the construction of the expression <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M213">View MathML</a> it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M214">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M23">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M216">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M25">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M6">View MathML</a>. The continuity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M207">View MathML</a> on Π follows from the continuity <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M213">View MathML</a> on each closed triangle Type A and Type B, and from the equality <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M214">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M23">View MathML</a>. By Remark 3.2 and from the condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M223">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M25">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M6">View MathML</a>, the continuity of the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M207">View MathML</a> on the closed rectangle <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M31">View MathML</a> follows. By virtue of (2.3) and (2.4) it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M62">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M63">View MathML</a> (see Theorem 3.1 in [15]). Then, on the basis of (3.6), (3.7), (3.9), (3.11) Theorem 2.2, Theorem 3.3 and (3.18), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M230">View MathML</a>

 □

4 An application of the matching operator in the Block-Grid method

Let G be an open simply connected staircase polygon, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M232">View MathML</a>, be its sides, including the ends, and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M233">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M234">View MathML</a>, be the interior angle formed by the sides <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M235">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M5">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M237">View MathML</a>). Furthermore, let s be the arc length measured along the boundary of G in the positive direction and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M17">View MathML</a> be the value of s at the vertex <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M10">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M240">View MathML</a> be a polar system of coordinates with pole in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M241">View MathML</a> and the angle <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M242">View MathML</a> taken counterclockwise from the side <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M5">View MathML</a>.

We consider the boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M244">View MathML</a>

(4.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M14">View MathML</a> are given functions, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M246">View MathML</a>

(4.2)

Moreover, at the vertices <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M241">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M248">View MathML</a> the conjugation conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M249">View MathML</a>

(4.3)

are satisfied. At the vertices <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M241">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M251">View MathML</a> no compatibility conditions for boundary functions are required; in particular the values of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M252">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M14">View MathML</a> at these vertices might be different. Additionally, it is required that when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M254">View MathML</a>, the boundary functions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M235">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M5">View MathML</a> are given as algebraic polynomials of arclength s measured along γ.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M257">View MathML</a>. We call the vertices <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M241">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M259">View MathML</a>, the singular vertices of the polygon G. We construct two fixed block sectors in the neighborhood of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M241">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M259">View MathML</a>, denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M262">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M263">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M264">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M265">View MathML</a>. On the closed sector <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M266">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M259">View MathML</a>, we consider the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M268">View MathML</a>, which has the following properties:

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M268">View MathML</a> is harmonic and bounded on the open sector <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M270">View MathML</a>;

(ii) continuous everywhere on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M266">View MathML</a> apart from the point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M241">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M259">View MathML</a> when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M274">View MathML</a>;

(iii) continuously differentiable on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M275">View MathML</a>;

(iv) satisfies the given boundary conditions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M276">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M277">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M259">View MathML</a>.

The function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M268">View MathML</a> with properties (i)-(iv) is given in [17].

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M280">View MathML</a>

(4.4)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M281">View MathML</a>

(4.5)

is the kernel of the Poisson integral for a unit circle.

The approximation of the integral representation given in the following lemma is used to construct an approximate solution of problem (4.1) around the singular vertices <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M241">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M259">View MathML</a>.

Lemma 4.1The solutionuof problem (2.1), (2.2) can be represented on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M284">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M259">View MathML</a>, in the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M286">View MathML</a>

(4.6)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M287">View MathML</a>is the curvilinear part of the boundary of the sector<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M288">View MathML</a>.

Proof The proof follows from Theorems 3.1 and 5.1 in [17]. □

We define the approximate solution in the whole polygon G by applying a version of the Block-Grid method introduced in [8] (see also [9]).

Let us consider, in addition to the sectors <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M270">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M288">View MathML</a>, the sectors <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M291">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M292">View MathML</a>, which are also in the neighborhood of each vertex <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M241">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M259">View MathML</a>, of the polygon G, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M295">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M296">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M297">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M298">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M299">View MathML</a>. Furthermore, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M300">View MathML</a>. We give the description of the Block-Grid method on a hexagonal grid:

(i) All singular corners <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M241">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M259">View MathML</a>, are separated by the double sectors <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M303">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M304">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M305">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M306">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M298">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M299">View MathML</a>. The polygon is covered by overlapping rectangles <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M309">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M310">View MathML</a>, and sectors <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M291">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M259">View MathML</a>, such that the distance from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M309">View MathML</a> to a singular point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M241">View MathML</a> is greater than <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M315">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M310">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M259">View MathML</a>.

(ii) On each rectangle <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M309">View MathML</a>, the seven point difference scheme for the approximation of Laplace’s equation on a hexagonal grid is used, with step size <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M319">View MathML</a>, h is a parameter, and as an approximate solution on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M320">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M259">View MathML</a>, the harmonic function (4.6) is used.

(iii) We use the matching operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M101">View MathML</a> constructed in Section 3 to connect the subsystems.

For obtaining the numerical solution of the algebraic system of equations (2.1), (2.2), we outline the procedure: Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M323">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M310">View MathML</a>, be certain fixed open rectangles with sides <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M325">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M326">View MathML</a> parallel to the sides of G, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M327">View MathML</a>. We use <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M328">View MathML</a> to denote the boundary of the rectangle <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M309">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M287">View MathML</a> is the curvilinear part of the boundary of the sector <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M288">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M332">View MathML</a>.

The overlapping condition is imposed on the arrangement of the rectangles <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M309">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M310">View MathML</a>: any point P lying on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M335">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M336">View MathML</a>, or located on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M337">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M259">View MathML</a>, falls inside at least one of the rectangles <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M339">View MathML</a>, where the distance from P to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M340">View MathML</a> is not less than some constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M341">View MathML</a> independent of P. The quantity <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M341">View MathML</a> is called the gluing depth of the rectangles <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M309">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M310">View MathML</a>.

We introduce the parameter <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M345">View MathML</a> and consider a hexagonal grid on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M309">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M310">View MathML</a>, with maximal possible step <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M348">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M349">View MathML</a> be the set of nodes on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M309">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M351">View MathML</a> be the set of nodes on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M328">View MathML</a>, and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M353">View MathML</a>. We denote the set of nodes on the closure of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M335">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M355">View MathML</a>, and the set of nodes on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M349">View MathML</a> whose distance from the boundary <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M335">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M358">View MathML</a> is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M32">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M360">View MathML</a>. We also have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M361">View MathML</a> denoting the set of nodes whose distance from the boundary <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M362">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M309">View MathML</a> is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M32">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M365">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M366">View MathML</a> be the set of nodes on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M367">View MathML</a>, and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M368">View MathML</a> be the set of remaining nodes on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M328">View MathML</a>. We also specify a natural number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M370">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M371">View MathML</a> is a fixed number and the quantities <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M372">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M373">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M374">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M259">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M376">View MathML</a>. On the arc <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M287">View MathML</a> we choose the points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M378">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M376">View MathML</a>, and denote the set of these points by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M380">View MathML</a>. Finally, let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M381">View MathML</a>

Consider the system of equations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M382">View MathML</a>

(4.7)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M383">View MathML</a>

(4.8)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M384">View MathML</a>

(4.9)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M385">View MathML</a>

(4.10)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M386">View MathML</a>

(4.11)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M387">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M259">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M389">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M390">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M391">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M392">View MathML</a> are defined as equations (2.9), (2.10), and (2.11) in Section 2, respectively.

The solution of the system of equations (4.7)-(4.11) is a numerical solution of problem (2.1), (2.2) on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M393">View MathML</a> (‘nonsingular’ part of the polygon G).

Theorem 4.2There is a natural number<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M394">View MathML</a>such that for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M395">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M396">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M341">View MathML</a>is the gluing depth, the system of equations (4.7)-(4.11) has a unique solution.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M398">View MathML</a> be a solution of the system of equations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M399">View MathML</a>

(4.12)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M400">View MathML</a>

(4.13)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M387">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M259">View MathML</a>. To prove the given theorem, we show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M403">View MathML</a>. On the basis of the structure of operators S and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M404">View MathML</a>, and the forms (3.5), (3.6), (3.7), (3.8), (3.9), and (3.10)-(3.11) of the matching operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M101">View MathML</a> and by the maximum principle (see Ch. 4, [11]) it follows that the nonzero maximum value of the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M398">View MathML</a> can be at the points on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M407">View MathML</a>. From estimation (2.29) in [18] the existence of the positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M394">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M409">View MathML</a> such that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M395">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M411">View MathML</a>

(4.14)

follows. However, taking (4.14) into account in (4.13) we have that the nonzero maximum value can not be at the points on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M407">View MathML</a> either. Since the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M413">View MathML</a> is connected, from equation (4.12) it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M403">View MathML</a>. □

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M51">View MathML</a> be the solution of the system of equations (4.7)-(4.11). The function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M416">View MathML</a>

(4.15)

is the approximation of the integral representation (4.6) with the use of the composite mid-point rule. We use the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M417','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M417">View MathML</a> as an approximate solution of problem (2.1), (2.2) on the closed block <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M320">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M259">View MathML</a> (‘singular’ parts of the polygon G).

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M420','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M420">View MathML</a>

(4.16)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M51">View MathML</a> is the solution of system (4.7)-(4.11) and u is the trace of the solution of (2.1), (2.2) on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M413">View MathML</a>. On the basis of (2.1), (2.2), (4.7)-(4.11), and (4.16), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M56">View MathML</a> satisfies the following difference equations:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M424','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M424">View MathML</a>

(4.17)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M425','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M425">View MathML</a>

(4.18)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M426">View MathML</a>

(4.19)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M427">View MathML</a>

(4.20)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M387">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M259">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M430">View MathML</a>

(4.21)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M431">View MathML</a>

(4.22)

Since all the rectangles <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M309">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M310">View MathML</a> are located away from the singular vertices <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M241">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M259">View MathML</a> of the polygon G, at a distance greater than <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M436">View MathML</a> independent of h, by virtue of the conditions (2.3) and (2.4), up to sixth order derivatives of the solution of problem (2.1), (2.2) are bounded on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M437','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M437">View MathML</a>. Then, by Taylor’s formula, from (4.21) we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M438">View MathML</a>

(4.23)

Furthermore, as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M439">View MathML</a> from (4.22) and Theorem 3.3, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M440','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M440">View MathML</a>

(4.24)

By analogy to the proof of Lemma 6.2 in [9], it is shown that there exists a natural number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M394">View MathML</a>, such that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M442">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M371">View MathML</a> being a fixed number,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M444','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M444">View MathML</a>

(4.25)

Theorem 4.3There exists a natural number<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M394">View MathML</a>such that for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M446','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M446">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M371">View MathML</a>being a fixed number,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M448">View MathML</a>

(4.26)

Proof The proof follows from estimations (4.23)-(4.25) and the principle of maximum by analogy to the proof of Theorem 6.1 in [9]. □

Theorem 4.4Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M51">View MathML</a>be the solution of the system of equations (4.7)-(4.11), and let an approximate solution of problem (2.1), (2.2) be found on blocks<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M320">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M259">View MathML</a>, by (4.15). There is a natural number<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M394">View MathML</a>such that for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M446','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M446">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M371">View MathML</a>being a fixed number, the following estimations hold:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M455','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M455">View MathML</a>

(4.27)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M456','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M456">View MathML</a>

(4.28)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M457','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M457">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M458','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M458">View MathML</a> .

Proof Estimation (4.27) is obtained from the integral representation (4.6) and formula (4.15) by using estimations (4.25) and (4.26). Estimation (4.28) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M458','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M458">View MathML</a> is obtained by using inequality (4.27) and Lemma 6.12 in [17]. □

5 Numerical results and discussion

To support the theoretical results, numerical examples have been solved in two different domains.

Example 5.1 Approximation in a rectangular domain.

Consider the rectangular domain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M460','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M460">View MathML</a>

with the boundary γ. The hexagonal grid (2.5), denoted <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M461','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M461">View MathML</a>, is assigned to the grid Π, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M462','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M462">View MathML</a> denotes the set of nodes on the boundary γ.

We consider the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M463','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M463">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M464','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M464">View MathML</a>

(5.1)

is the exact solution in the rectangular domain.

This example is solved using the incomplete LU-decomposition method (see [19], Ch. 5), and all the calculations are carried out in double precision. As a convergence test, we request the maximum residual error to be 10−12, and as a starting point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M465','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M465">View MathML</a> is used.

Table 1 gives the values obtained in the maximum norm of the difference between the exact and the approximate solutions, for the values of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M466','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M466">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M467','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M467">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M468','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M468">View MathML</a>. The ratios <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M469','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M469">View MathML</a> have also been included, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M3">View MathML</a> order of accuracy corresponds to 24 of the value <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M471','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M471">View MathML</a>.

Table 1. Approximation in a rectangle with smooth exact solution

Example 5.2 Less smooth function.

We consider the same problem as in Example 5.1 with the exact solution

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M474','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M474">View MathML</a>

(5.2)

which is less smooth than (5.1). The results obtained are consistent with the theoretical results and are summarized in Table 2.

Table 2. Approximation in a rectangle with less smooth exact solution

Example 5.3 The matching operator.

Examples of the matching operator have also been considered in the domain Π. The coordinate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M477','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M477">View MathML</a> is chosen, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M478','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M478">View MathML</a>. The harmonic function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M479','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M479">View MathML</a>

(5.3)

is assumed to be the exact solution. The result in Table 3 is obtained using <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M480','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M480">View MathML</a> and demonstrates high accuracy of the above constructed matching operator.

Table 3. Results for approximation of inner points with the matching operator

The second coordinate considered demonstrates the accuracy of the approximation of near-boundary points. The point chosen is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M483','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M483">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M484','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M484">View MathML</a>, and equation (3.13) is used for approximation. Again, the harmonic function (5.3) is used as the exact solution. Lastly, a point near one of the corners of the domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M485','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M485">View MathML</a> has been considered, where the nodes of evaluation emerge outside of the domain from both adjacent sides of the corner. The function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M486','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M486">View MathML</a>

is used as the exact solution. The results obtained are summarized in Table 4.

Table 4. Results for approximation of near boundary points with the matching operator

Example 5.4 Approximation in an L-shaped domain.

The final example is solved in an L-shaped domain with an angle singularity at the origin, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M489','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M489">View MathML</a>. The domain is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M490','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M490">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M491','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M491">View MathML</a>, and is covered by four overlapping rectangles and a sector. The singular part is defined to be the region

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M492','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M492">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M493','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M493">View MathML</a>, and the nonsingular part is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M494','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M494">View MathML</a>. The system of Block-Grid equations is solved by Schwarz’s alternating method. The quadrature nodes on the circular arc, whose radius is taken as 0.75, and the overlapping boundaries of the rectangles are renewed after each Schwarz’s iteration. The nodes on the circular arc, the inner boundaries of the overlapping rectangles and the nodes in the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M495','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M495">View MathML</a> are renewed using the matching operator constructed above. Since the boundary functions are harmonic polynomials on the sides <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M496','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M496">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M497','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M497">View MathML</a>, the nodes whose neighbors emerge outside of the domain from these sides are approximated using the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M498','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M498">View MathML</a>. Finally, the solution on the singular part is approximated using the integral representation [3,9].

The problem considered is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M499','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M499">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M500','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M500">View MathML</a>

is the exact solution. Accordingly, the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M501','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M501">View MathML</a> used in the integral representation is constructed as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M502','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M502">View MathML</a>

The results in Tables 5 and 6 show the solution for different pairs <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M503','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M503">View MathML</a>, where N is the number of quadrature nodes, h is the mesh size of the hexagonal grid.

Table 5. The order of convergence in ‘nonsingular’ part

Table 6. The order of convergence in ‘singular’ part

6 Conclusions

The fourth order classical 7-point scheme on a hexagonal grid is applied on a rectangular domain. This leads to some of the nodes emerging from the two parallel sides while approximating points whose distance from the boundary is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/73/mathml/M32">View MathML</a>. This problem was overcome by devising an approximating equation for near-boundary nodes, which included the use of three inner nodes around the point of evaluation and three points lying on the boundary. The fourth order matching operator has been constructed on the hexagonal grid functions. It is applied to construct a fourth order accurate interpolating function, on the closed rectangle, for the numerical solution of Laplace’s equation on the hexagonal grids. Further, the matching operator and the hexagonal grid approximation in a rectangle are used to obtain and justify the Block-Grid method in solving the Dirichlet problem for Laplace’s equation on staircase polygons.

Numerical examples have been provided as an illustration of the theoretical results mentioned above.

The matching operator constructed can be applied to many other forms of domain decomposition or combined methods. It will also be an interesting study to extend the approximation of the methods to using mixed or Neumann boundary conditions.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

References

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