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A BDDC algorithm for the mortar-type rotated Q1 FEM for elliptic problems with discontinuous coefficients

Yaqin Jiang1 and Jinru Chen2

Author Affiliations

1 School of Sciences, Nanjing University of Posts and Telecommunications, Nanjing, 210046, P.R. China

2 Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210046, P.R. China

Boundary Value Problems 2014, 2014:79  doi:10.1186/1687-2770-2014-79


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/79


Received:14 January 2014
Accepted:17 March 2014
Published:3 April 2014

© 2014 Jiang and Chen; 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

In this paper, we propose a BDDC preconditioner for the mortar-type rotated <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M1">View MathML</a> finite element method for second order elliptic partial differential equations with piecewise but discontinuous coefficients. We construct an auxiliary discrete space and build our algorithm on an equivalent auxiliary problem, and we present the BDDC preconditioner based on this constructed discrete space. Meanwhile, in the framework of the standard additive Schwarz methods, we describe this method by a complete variational form. We show that our method has a quasi-optimal convergence behavior, i.e., the condition number of the preconditioned problem is independent of the jumps of the coefficients, and depends only logarithmically on the ratio between the subdomain size and the mesh size. Numerical experiments are presented to confirm our theoretical analysis.

MSC: 65N55, 65N30.

Keywords:
domain decomposition; BDDC algorithm; mortar; rotated <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M1">View MathML</a> element; preconditioner

1 Introduction

The method of balancing domain decomposition by constraints (BDDC) was first introduced by Dohrmann in [1]. Mandel and Dohrmann restated the method in an abstract manner, and provided its convergence theory in [2]. The BDDC method is closely related to the dual-primal FETI (FETI-DP) method [3], which is one of dual iterative substructuring methods. Each BDDC and FETI-DP method is defined in terms of a set of primal continuity. The primal continuity is enforced across the interface between the subdomains and provides a coarse space component of the preconditioner. In [4], Mandel, Dohrmann, and Tezaur analyzed the relation between the two methods and established the corresponding theory.

In the last decades, the two methods have been widely analyzed and successfully been extended to many different types of partial differential equations. In [3], the two algorithms for elliptic problems were rederived and a brief proof of the main result was given. A BDDC algorithm for mortar finite element was developed in [5], meanwhile, the author also extended the FETI-DP algorithm to elasticity problems and Stokes problems in [6,7], respectively. These algorithms are based on locally conforming finite element methods, and the coarse space components of the algorithms are related to the cross-points (i.e., corners), which are often noteworthy points in domain decomposition methods (DDMs). Since the cross-points are related to more than two subregions, thus it is not convenient to design the domain decomposition algorithm.

The BDDC method derives from the Neumann-Neumann domain decomposition method (see [8]). The difference is that the BDDC method applies an additive rather than a multiplicative coarse grid correction, and substructure spaces have some constraints which result in non-singular subproblems. Thus we need not modify the bilinear forms on subdomains, and we can solve each subproblem and coarse problem in parallel.

The rotated <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M1">View MathML</a> element is an important nonconforming element. It was introduced by Rannacher and Turek in [9] for stokes equations originally, and it is the simplest example of a divergence-stable nonconforming element on quadrilaterals. Since its degree of freedom is integral average on element edge which is not related to the corners, and each degree of freedom on subdomain interfaces is only included in two neighboring subdomains, so it is easy to design the BDDC algorithm.

The mortar technique was introduced in [10]. This method is nonconforming domain decomposition methods with nonoverlapping subdomains. The meshes on different subdomains need not align across subdomain interfaces, and the matching of discretizations on adjacent subdomains is only enforced weakly. This offers the advantages of freely choosing highly varying mesh sizes on different subdomains and is very promising to approximate the problems with abruptly changing diffusion coefficients or local anisotropic.

In this paper, we study the BDDC algorithm for the mortar-type rotated <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M1">View MathML</a> element for the second order elliptic problem with discontinuous coefficients, where the discontinuities lie only along the subdomain interfaces. Following the technique in [11], we construct an auxiliary discrete space and build our BDDC algorithm on an equivalent auxiliary problem. This approach overcomes the difficulty caused by the mortar condition and simplifies the implementation of the BDDC preconditioning iteration. Furthermore, since the rotated <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M1">View MathML</a> element is not related to the subdomain’s vertices, we can complete our theoretical analysis conveniently. It is proved that the condition number of the preconditioned operator is independent of the jumps of the coefficients and only depends logarithmically on the ratio between the subdomain size and mesh size. Numerical experiments are presented to confirm our theoretical analysis.

The rest of this paper is organized as follows: in Section 2, we introduce the model problem and the auxiliary problem. Section 3 gives the BDDC algorithm and proposes the BDDC preconditioner. Several technical tools are presented and analyzed in Section 4. In Section 5, we give the proof of the main result. Last section provides numerical experiments. For convenience, the symbols ⪯, ⪰ and ≍ are used, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M8">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M9">View MathML</a> mean that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M10">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M11">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M12">View MathML</a> for some constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M13">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M14">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M15">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M16">View MathML</a> that are independent of discontinuous coefficients and mesh size.

2 Preliminaries

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M17">View MathML</a> be a bounded, simply connect rectangular or L-shaped domain, we divide Ω into several nonoverlapping regular rectangular subdomains <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M18">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M19">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M20">View MathML</a>. Consider the following model problem: Find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M21">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M22">View MathML</a>

(2.1)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M23">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M24">View MathML</a>, the coefficients <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M25">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M26">View MathML</a>) are piecewise positive constants over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M18">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M28">View MathML</a>).

For simplicity, we only consider the geometrically conforming case, i.e., the intersection between the closure of two different subdomains is empty, or a vertex, or an edge. The subdomains <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M29">View MathML</a> together form a coarse partition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M30">View MathML</a>, we denote the diameter of each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M18">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M32">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M33">View MathML</a> be a quasi-uniform partition with the mesh size <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M34">View MathML</a>, made up of shape regular rectangles in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M18">View MathML</a>. The resulted partition can be nonmatched across adjacent subdomain interfaces. We denote the sets of edges of the triangulation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M33">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M18">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M38">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M39">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M40">View MathML</a> respectively, and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M41">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M42">View MathML</a> be the sets of vertices of the triangulation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M43">View MathML</a> that are in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M44">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M45">View MathML</a> respectively.

For each triangulation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M43">View MathML</a>, the rotated <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M1">View MathML</a> element space is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M48">View MathML</a>

Let the global discrete space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M49">View MathML</a>. We equip the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M50">View MathML</a> with the following seminorm:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M51">View MathML</a>

We denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M52">View MathML</a> the common open edge of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M18">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M54">View MathML</a>, and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M55">View MathML</a>. Each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M52">View MathML</a> can be regarded as two sides corresponding to the two subdomains <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M18">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M54">View MathML</a>. We define one of the sides of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M52">View MathML</a> as mortar denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M60">View MathML</a> and the other one as nonmortar denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M61">View MathML</a>, here m represents the indexing of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M52">View MathML</a> (see Figure 1). We assume that: (1) the mortar for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M63">View MathML</a> is chosen by the condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M64">View MathML</a>; (2) there is at least one subdomain which has two mortar sides associated with each cross point; (3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M65">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M66">View MathML</a> is bounded. The first condition used in choosing mortar sides is essential (see the numerical tests in [12]). The last condition is technical but not essential for the convergence analysis. Along each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M52">View MathML</a>, there are two independent and different 1-D meshes which are denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M68">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M69">View MathML</a>. For each nonmortar side <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M70">View MathML</a>, we denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M71">View MathML</a> an auxiliary test space whose functions are piecewise constant on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M69">View MathML</a>. We denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M73">View MathML</a> the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M74">View MathML</a>-orthogonal projection from the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M75">View MathML</a> space to the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M76">View MathML</a> space.

thumbnailFigure 1. Nonmatching grid.

Now we define the mortar-type rotated <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M1">View MathML</a> space as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M78">View MathML</a>

(2.2)

here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M79">View MathML</a> is the restriction of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M80">View MathML</a> to the mortar side <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M60">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M82">View MathML</a> is the restriction of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M83">View MathML</a> to the nonmortar side <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M61">View MathML</a>. The condition in (2.2) for each interface is called mortar condition. The mortar-type rotated <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M1">View MathML</a> element approximation of problem (2.1) is: find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M86">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M87">View MathML</a>

(2.3)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M88">View MathML</a>

It can easily be shown that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M89">View MathML</a> is positive definite on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M90">View MathML</a>, which yields the existence and uniqueness of the discrete solution. The error estimate between the discrete and the continuous solution is discussed in [13].

Since the mortar condition depends on both the degrees of freedom on the interfaces and the ones near the interfaces, it is difficult to construct a preconditioner directly for (2.3). To overcome this difficulty, we introduce a new discrete space and an auxiliary problem which is equivalent to problem (2.3).

For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M91">View MathML</a>, we define an element <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M92">View MathML</a> that satisfies the following conditions:

• for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M93">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M94">View MathML</a>

(2.4)

• for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M95">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M96">View MathML</a>

(2.5)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M97">View MathML</a> is a piecewise constant function on elements of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M68">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M99">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M100">View MathML</a>. Note that the average value of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M101">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M102">View MathML</a> can be calculated by (2.5).

By the above definition, all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M101">View MathML</a> associated with v form a space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M104">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M105">View MathML</a>

For the two related spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M90">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M107">View MathML</a>, we have the following result.

Lemma 2.1 ([12])

For any pair of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M108">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M109">View MathML</a>defined above, the following is true:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M110">View MathML</a>

(2.6)

Now we introduce the auxiliary problem, that is, to find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M111">View MathML</a> which satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M112">View MathML</a>

(2.7)

Define an operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M113">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M114">View MathML</a>

From the above lemma, we only need to construct a preconditioner for the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M115">View MathML</a>.

3 BDDC algorithm

In this section, we introduce our BDDC preconditioner for problem (2.7) and describe the BDDC algorithm.

We first define a discrete harmonic operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M116">View MathML</a> associated with the rotated <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M1">View MathML</a> element: for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M118">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M119">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M120">View MathML</a>

here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M121">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M122">View MathML</a>. We define ℋ as a corresponding piecewise harmonic operator on the auxiliary space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M123">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M124">View MathML</a>.

In order to introduce our domain decomposition method, we decompose the auxiliary discrete space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M123">View MathML</a> as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M126">View MathML</a>

(3.1)

where the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M127">View MathML</a> is a piecewise harmonic function space defined as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M128">View MathML</a>

We define a space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M129">View MathML</a>. The space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M130">View MathML</a> is between <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M131">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M132">View MathML</a>, and our BDDC preconditioner is mainly constructed on this space.

As we know, the technical aspect in DDMs is that the preconditioner includes a coarse problem which can enhance the convergence. In view of the characteristic of the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M130">View MathML</a>, we select the standard coarse space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M134">View MathML</a> which is the rotated <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M1">View MathML</a> finite element space associated with the coarse partition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M30">View MathML</a>, and it satisfies primal constraints on subdomain interfaces.

The substructure space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M137">View MathML</a> with constraints is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M138">View MathML</a>

Denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M139">View MathML</a>. The coarse space and the product space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M140">View MathML</a> play an important role in the description and analysis of our iterative method.

To present our BDDC preconditioner, we introduce several space transfer operators. Define an interpolation operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M141">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M142">View MathML</a>

The intergrid transfer operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M143">View MathML</a> is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M144">View MathML</a>

Define an extension operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M145">View MathML</a> as

• for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M146">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M147">View MathML</a>;

• for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M148">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M149">View MathML</a>;

• for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M150">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M151">View MathML</a> satisfies (2.5).

Its transpose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M152">View MathML</a> is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M153">View MathML</a>

Denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M154">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M155">View MathML</a>, the corresponding transpose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M156">View MathML</a> is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M157">View MathML</a>

We also need to define another prolongation operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M158">View MathML</a> as follows:

• if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M159">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M160">View MathML</a>;

• if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M100">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M162">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M163">View MathML</a>;

• if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M164">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M165">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M166">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M167">View MathML</a>;

• if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M168">View MathML</a>, it follows from (2.5) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M169">View MathML</a> can be obtained by the edge average values on associated mortar sides;

• else, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M167">View MathML</a>.

In what follows, we describe our BDDC preconditioning algorithm, we apply the basic framework of additive Schwarz method (or parallel subspace correction method [14]). From the decomposition (3.1), we only need to choose appropriate subspace solvers.

First of all, the coarse subspace solver <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M171">View MathML</a> is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M172">View MathML</a>

On each subdomain, similar operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M173">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M174">View MathML</a> are defined, respectively, by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M175">View MathML</a>

Remark 3.1 The bilinear form on the coarse space can be different from that on substructure space, here we only use the exact solvers. On each subdomain, we avoid the possible singularity of local subproblem and we need not modify the bilinear forms.

Now we define our BDDC preconditioner as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M176">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M177">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M178">View MathML</a> is the corresponding transpose defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M179">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M180">View MathML</a> be an operator from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M127">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M134">View MathML</a> defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M183">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M184">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M185">View MathML</a> be the operators from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M127">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M137">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M188">View MathML</a> defined, respectively, by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M189">View MathML</a>

Then the BDDC preconditioned operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M190">View MathML</a> can be written as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M191">View MathML</a>

We have the following main result.

Theorem 3.1The BDDC preconditioned operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M192">View MathML</a>satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M193">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M194">View MathML</a>.

4 Technical tools

In this section we state and prove a few technical lemmas necessary for the proof of Theorem 3.1. Our theoretical analysis is based on the substructuring theory of conforming elements.

We assume <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M195">View MathML</a> be the bilinear conforming element space associated with the partition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M43">View MathML</a>. We split the interface <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M197">View MathML</a> into four open edges ℰ, and define a restriction operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M198">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M199">View MathML</a>) as: for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M200">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M201">View MathML</a>

For the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M202">View MathML</a>, we have the following result.

Lemma 4.1 ([15])

For an edgeof<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M197">View MathML</a>, then for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M200">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M205">View MathML</a>

Remark 4.1 The above lemma is related to vertex-edge-face arguments in substructuring methods, in view of the characteristic for the rotated <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M1">View MathML</a> element, here the results only concern the inequalities for faces.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M207">View MathML</a> be the conforming element space of bilinear continuous functions on the partition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M208">View MathML</a> which is constructed by joining the midpoints of the edges of elements of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M43">View MathML</a>. We now introduce a local equivalence map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M210">View MathML</a> as follows (cf.[13]).

Definition 4.2 Given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M211">View MathML</a>, we define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M212">View MathML</a> by the values of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M213">View MathML</a> at the vertices of the partition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M208">View MathML</a>.

• If P is a central point of E, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M215">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M216">View MathML</a>

• If P is a midpoint of one edge <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M217">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M215">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M219">View MathML</a>

• If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M220">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M221">View MathML</a>

where the sum is taken over all edges <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M222">View MathML</a> with the common vertex P, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M223">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M224">View MathML</a>.

• If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M225">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M226">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M227">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M228">View MathML</a> are the left and right neighbor edges of P, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M229">View MathML</a>. If P is a vertex of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M18">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M231">View MathML</a>.

Define the pseudo-inverse map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M232">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M233">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M234">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M235">View MathML</a>, P is the midpoint of e. Obviously, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M236">View MathML</a>

For the operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M237">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M238">View MathML</a>, we have the following results (see [13]):

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M239">View MathML</a>

(4.1)

Lemma 4.3For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M240">View MathML</a>, we can split<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M241">View MathML</a>into<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M242">View MathML</a>, and we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M243">View MathML</a>

(4.2)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M244">View MathML</a>, and for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M245">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M246">View MathML</a>; for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M247">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M248">View MathML</a>.

Proof By (4.1), Lemma 4.1, the inverse trace theorem, the trace theorem, and the Poincaré inequality, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M249">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M250">View MathML</a> is a piecewise bilinear conforming element harmonic operator, and we have used the minimal energy property of discrete harmonic functions. □

5 Proof of Theorem 3.1

In the proof of Theorem 3.1 we use the abstract framework of ASM methods (see [16]), we need to prove three assumptions. Assumption II follows from the standard coloring argument, we only need to prove Assumption I and Assumption III.

First we show the following stability of the decomposition.

Lemma 5.1 (Assumption I)

For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M251">View MathML</a>, we have the following decomposition:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M252">View MathML</a>

(5.1)

which satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M253">View MathML</a>

(5.2)

Proof First we show the decomposition (5.1). For any function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M254">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M255">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M256">View MathML</a>, obviously <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M257">View MathML</a> is a piecewise discrete harmonic function. So we denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M258">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M259">View MathML</a>. From the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M260">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M261">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M262">View MathML</a>

and by the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M263">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M264">View MathML</a>

where we have used the fact <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M265">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M266">View MathML</a>. Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M267">View MathML</a> and the equality (5.1) holds.

Now we prove the stability of decomposition (5.2). Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M268">View MathML</a>. Using Lemma 3.5 in [12], Poincaré-Friedrichs’ inequality and scaling argument, we derive

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M269">View MathML</a>

(5.3)

From (5.3) and the discrete equivalent norm, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M270">View MathML</a>

(5.4)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M185">View MathML</a> is an orthogonal projection with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M272">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M273">View MathML</a>

(5.5)

Meanwhile, from the fact that the harmonic function has minimal energy norm and (5.4)-(5.5), we deduce

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M274">View MathML</a>

(5.6)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M275">View MathML</a>

(5.7)

So (5.4)-(5.6) lead to (5.2). □

Next we state the local stability as follows.

Lemma 5.2 (Assumption III)

For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M276">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M277">View MathML</a>

(5.8)

For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M278">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M279">View MathML</a>

(5.9)

Proof To prove (5.8) we first introduce a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M280">View MathML</a> associated with a mortar side <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M281">View MathML</a>, which satisfies the following:

• for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M282">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M283">View MathML</a>;

• for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M284">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M285">View MathML</a>;

• for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M150">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M287">View MathML</a> satisfies (2.5).

Then we can decompose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M288">View MathML</a> as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M289">View MathML</a>

(5.10)

here we have used the fact that the degrees of freedom on the interface Γ of the function u are as same as that of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M290">View MathML</a>, and the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M291">View MathML</a> is defined by the average values on the edge elements, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M292">View MathML</a>

Note that the support of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M293">View MathML</a> is on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M294">View MathML</a>, and using Lemma 3.4 in [12] we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M295">View MathML</a>

(5.11)

Since the degrees of freedom on the interface <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M197">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M297">View MathML</a> are only nonzero on the edge <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M60">View MathML</a>, using Lemma 4.3, we deduce

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M299">View MathML</a>

(5.12)

From (5.10)-(5.12), we complete the proof of (5.8).

Using similar techniques to those in (5.8), and summing over all subdomains, we can complete the proof of (5.9). □

6 Numerical results

In this section, we show numerical results of our method using the model problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M300">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M301">View MathML</a>. The domain is composed of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M302">View MathML</a> sub-squares, their mesh sizes are H, and the sub-squares are divided into smaller ones with mesh sizes <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M303">View MathML</a> in mortar subdomains; and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M304">View MathML</a> in nonmortar subdomains. The coefficient ρ is either 1 or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M305">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M306">View MathML</a>).

We use the preconditioned conjugate gradient (PCG) method with zero initial guess for the discrete system of equations. The stopping criterion for the PCG method is when the 2-norm of the residual is reduced by the factor of 10−6 of the initial guess. An estimate for the condition number of the corresponding system is computed by using the Lanczos algorithm.

In Table 1, we show the number of iterations and the condition numbers with different ratio <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M307">View MathML</a>. In Figure 2, we plot the condition number as the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M308">View MathML</a> for 16 domains. From the results in Table 1 and Figure 2, we see that the convergence of our method is quasi-optimal since the number of iterations is independent of the jumps of the coefficients, and almost independent of the mesh size.

thumbnailFigure 2. Plot of the condition numbers as the function of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M309">View MathML</a>.

Table 1. The number of iterations and condition numbers for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/79/mathml/M310">View MathML</a>

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All results belong to YJ and JC. All authors read and approved the final manuscript.

Acknowledgements

The work was supported by the National Natural Science Foundation of China (Grant Nos. 11371199 and 11301275), Jiangsu Provincial 2011 Program (Collaborative Innovation Center of Climate Change), the Program of Natural Science Research of Jiangsu Higher Education Institutions of China (Grant No. 12KJB110013), the Doctoral fund of Ministry of Education of China (Grant No. 20123207120001), and Jiangsu Key Lab for NSLSCS (Grant No. 201306). Moreover the authors are grateful to anonymous referees for their constructive comments and suggestions.

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