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
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  and , 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 , the fourth order matching operator is constructed and used for the fourth order composite grids and in  and  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 .
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 on a hexagonal grid, the six equidistant nodes surrounding are used, and the truncation error obtained is . 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 ). 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
We consider the system of finite difference equations
From formulae (2.9) and (2.10) it follows that the coefficients of the expressions and 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 ).
are satisfied, then
Proof The proof of this lemma is similar to the proof of the comparison theorem (see Ch. 4 in ). □
are the truncation errors of equations (2.6) and (2.7), respectively.
We represent the solution of (2.13)-(2.15) as
On the basis of the maximum principle, we obtain
From (2.19), (2.20), and (2.21) it follows that
3 Construction of the fourth order matching operator
By analogy with the idea used in , we construct the operator from the condition that the expression
Let denote the set of points such that all the nodes to determine the expression belong to , and contain the points P, where some of the nodes emerge through the side , . We construct the fourth order matching operator by considering the cases when the point P belongs to one of the sets or .
By solving system (3.3) and rearranging (3.2) for u, we obtain the equation
We now take into consideration the nodes and which are symmetric to the points and , respectively, with respect to the x-axis. Since , for , and odd with respect to y, and , , is even with respect to y, from (3.4) we have
It is easy to check that
Again, we place the origin of the rectangular system of coordinates at the node and direct the positive axis of x along the grid line, so that P has the coordinates , where . We form an artificial grid by taking the following points:
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
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.
Figure 1. Shapes of triangles in a hexagon.
We consider triangles of Type A with . The nodes used in are shown in Figure 2.
Figure 2. Type A triangle with.
In the case , , the 17 nodes used have the same distribution as the reflection of the nodes in Figure 2 about the line . In the cases , and , the nodes for 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.
We position the origin of the rectangular system of coordinates on so that the point P lies on the positive yaxis, and the xaxis is in the direction of the vertex along . It is obvious that if , where . Hence, when the function , , is represented using Taylor’s formula about the point in the neighborhood of the origin, we define , , of (3.1) as
for , and keeping in mind that is odd extendable, we complete the definition with for . Clearly, in the given neighborhood, is equal to the harmonic polynomial , with an accuracy of . To form an expression for the matching operator , we use
Hence using (3.10) or (3.11), with the addition of the term
whereuis the exact solution of problem (2.1), (2.2).
Proof According to Theorem 3.1 in , from conditions (3.15) and (3.16) it follows that . Then on the basis of (3.1), (3.5), (3.8), (3.13), and Remark 3.2, we obtain inequality (3.17). □
whereuis the solution of problem (2.1), (2.2).
Proof From the construction of the expression it follows that on , and on , . The continuity of on Π follows from the continuity on each closed triangle Type A and Type B, and from the equality on . By Remark 3.2 and from the condition on , , the continuity of the function on the closed rectangle follows. By virtue of (2.3) and (2.4) it follows that , (see Theorem 3.1 in ). Then, on the basis of (3.6), (3.7), (3.9), (3.11) Theorem 2.2, Theorem 3.3 and (3.18), we obtain
4 An application of the matching operator in the Block-Grid method
Let G be an open simply connected staircase polygon, let , , be its sides, including the ends, and let , , be the interior angle formed by the sides and (). Furthermore, let s be the arc length measured along the boundary of G in the positive direction and be the value of s at the vertex , be a polar system of coordinates with pole in and the angle taken counterclockwise from the side .
We consider the boundary value problem
are satisfied. At the vertices for no compatibility conditions for boundary functions are required; in particular the values of and at these vertices might be different. Additionally, it is required that when , the boundary functions on and are given as algebraic polynomials of arclength s measured along γ.
Let . We call the vertices , , the singular vertices of the polygon G. We construct two fixed block sectors in the neighborhood of , , denoted by , , where , . On the closed sector , , we consider the function , which has the following properties:
The function with properties (i)-(iv) is given in .
is the kernel of the Poisson integral for a unit circle.
Proof The proof follows from Theorems 3.1 and 5.1 in . □
Let us consider, in addition to the sectors , , the sectors and , which are also in the neighborhood of each vertex , , of the polygon G, with , and , , where . Furthermore, let . We give the description of the Block-Grid method on a hexagonal grid:
(i) All singular corners , , are separated by the double sectors , , with , , and . The polygon is covered by overlapping rectangles , , and sectors , , such that the distance from to a singular point is greater than for all and .
(ii) On each rectangle , the seven point difference scheme for the approximation of Laplace’s equation on a hexagonal grid is used, with step size , h is a parameter, and as an approximate solution on , , the harmonic function (4.6) is used.
For obtaining the numerical solution of the algebraic system of equations (2.1), (2.2), we outline the procedure: Let , , be certain fixed open rectangles with sides and parallel to the sides of G, and . We use to denote the boundary of the rectangle , is the curvilinear part of the boundary of the sector and .
The overlapping condition is imposed on the arrangement of the rectangles , : any point P lying on , , or located on , , falls inside at least one of the rectangles , where the distance from P to is not less than some constant independent of P. The quantity is called the gluing depth of the rectangles , .
We introduce the parameter and consider a hexagonal grid on , , with maximal possible step . Let be the set of nodes on , let be the set of nodes on , and let . We denote the set of nodes on the closure of by , and the set of nodes on whose distance from the boundary of is by . We also have denoting the set of nodes whose distance from the boundary of is and . Let be the set of nodes on , and let be the set of remaining nodes on . We also specify a natural number , where is a fixed number and the quantities , , and , , . On the arc we choose the points , , and denote the set of these points by . Finally, let
Consider the system of equations
where , . To prove the given theorem, we show that . On the basis of the structure of operators S and , and the forms (3.5), (3.6), (3.7), (3.8), (3.9), and (3.10)-(3.11) of the matching operator and by the maximum principle (see Ch. 4, ) it follows that the nonzero maximum value of the function can be at the points on . From estimation (2.29) in  the existence of the positive constants and such that for
is the approximation of the integral representation (4.6) with the use of the composite mid-point rule. We use the function as an approximate solution of problem (2.1), (2.2) on the closed block , (‘singular’ parts of the polygon G).
where is the solution of system (4.7)-(4.11) and u is the trace of the solution of (2.1), (2.2) on . On the basis of (2.1), (2.2), (4.7)-(4.11), and (4.16), satisfies the following difference equations:
Since all the rectangles , are located away from the singular vertices , of the polygon G, at a distance greater than 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 . Then, by Taylor’s formula, from (4.21) we obtain
By analogy to the proof of Lemma 6.2 in , it is shown that there exists a natural number , such that for all , being a fixed number,
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 . □
Theorem 4.4Letbe 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, , by (4.15). There is a natural numbersuch that for all, being a fixed number, the following estimations hold:
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 is obtained by using inequality (4.27) and Lemma 6.12 in . □
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
We consider the problem
is the exact solution in the rectangular domain.
This example is solved using the incomplete LU-decomposition method (see , 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 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 , , i.e., . The ratios have also been included, where order of accuracy corresponds to 24 of the value .
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
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.
is assumed to be the exact solution. The result in Table 3 is obtained using 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 , where , 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 has been considered, where the nodes of evaluation emerge outside of the domain from both adjacent sides of the corner. The function
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.
where , and the nonsingular part is . 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 are renewed using the matching operator constructed above. Since the boundary functions are harmonic polynomials on the sides and , the nodes whose neighbors emerge outside of the domain from these sides are approximated using the function . Finally, the solution on the singular part is approximated using the integral representation [3,9].
The problem considered is
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 . 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.
The authors declare that they have no competing interests.
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
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