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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 948564, 8 pages
http://dx.doi.org/10.1155/2013/948564
Research Article

Analysis of the Block-Grid Method for the Solution of Laplace's Equation on Polygons with a Slit

Department of Mathematics, Eastern Mediterranean University, Gazimagusa, Cyprus, 095 Mersin 10, Turkey

Received 22 April 2013; Accepted 30 April 2013

Academic Editor: Allaberen Ashyralyev

Copyright © 2013 S. Cival Buranay. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The error estimates obtained for solving Laplace's boundary value problem on polygons by the block-grid method contain constants that are difficult to calculate accurately. Therefore, the experimental analysis of the method could be essential. The real characteristics of the block-grid method for solving Laplace's equation on polygons with a slit are analysed by experimental investigations. The numerical results obtained show that the order of convergence of the approximate solution is the same as in the case of a smooth solution. To illustrate the singular behaviour around the singular point, the shape of the highly accurate approximate solution and the figures of its partial derivatives up to second order are given in the “singular” part of the domain. Finally a highly accurate formula is given to calculate the stress intensity factor, which is an important quantity in fracture mechanics.

1. Introduction

In the past few decades, in order to improve the accuracy and resolve the convergence difficulties that appear in the neighbourhood of singular points, many different methods have been proposed for the numerical solution of plane elliptic boundary value problems with singularities. Among many approaches, a special emphasis has been placed on the construction of combined methods, in which differential properties of the solution in different parts of the domain are used (see [1]).

In [26] a new combined difference-analytical method called the block-grid method (BGM) is given for solving the Laplace equation on polygons, when the boundary functions on the sides causing the singular vertices are given as algebraic polynomials of the arclength. This method is a combination of the exponentially convergent block method (see [7, 8]) in the “singular” part, and the finite difference method, which has a simple structure on the “nonsingular” part of the polygon. A th order gluing operator is constructed for gluing together the grids and the blocks. The uniform estimate of the error of the BGM is of order ( is the mesh step) when the given boundary function on the boundary of the “nonsingular” part might be from the Hölder classes , (see [24] for , [6] for , and [5] for ). For the errors of -order derivatives () the estimation is obtained in a finite neighborhood of the vertices, where is the distance from the current point to the vertex in question, and is the value of the interior angle at the considered vertex. Moreover, BGM can give a simple and highly accurate formula for the stress intensity factor which is an important quantity from an engineering standpoint.

The experimental investigation of the block-grid method is important and numerical results could be interesting to support the theoretical results in [26]. The objective of this paper is to analyze the real characteristics of the BGM for solving the Laplace equation on polygons with a slit. For this purpose a slit problem on a square domain whose exact solution is known is considered. The computational algorithm by the BGM with 5-point and 9-point schemes is given and implemented. The obtained numerical results justify the theoretical results given in [25]. Moreover, for the approximate solution (by 9-point scheme with ) and the error function the graphs are given to demonstrate the high accuracy of the block-grid method. The shapes of the partial derivatives , , , , are given to illustrate the singular behavior in the “singular” part of the domain. Furthermore, a simple and highly accurate formula is given to calculate the stress intensity factor.

The experimental analyses of the different methods on slit problems were given in many papers (see [9, 10]).

2. The Slit Problem and the Integral Representation of the Solution

Let be an open domain in the plane , that is obtained from the unit square by making a cut along the positive semiaxis from the center (see Figure 1). Let , , be its sides, including the ends, enumerated counterclockwise, , , be the boundary of , is the interior angle formed by the sides and . Denote by the vertex of this angle and let be a polar system of coordinates with a pole in , where the angle is taken counterclockwise from the side .

948564.fig.001
Figure 1: Covering the square domain with a slit by overlapping rectangles and sector.

We consider the boundary value problem where and is the value of the function on .

In the neighborhood of , we construct two fixed block-sectors , , where , .

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

Lemma 1. The solution of the boundary value problem (1), (2) can be represented on , in the form where is the curvilinear part of the boundary of .

Proof. The proof follows from Theorem 3.1 in [8] by taking into account that .

3. The Block-Grid Method for the Slit Problem

The realization of the BGM for the solution of the problem (1), (2) is as follows. Let and be a polygonal line which lies on with a positive distance from the vertex and from the curvilinear boundary of . The set of points from up to is denoted by which is called the “singular” part of and the set is the “nonsingular” part of . In addition to the sector in the neighborhood of the vertex of the polygon we construct two more sectors and . Let and , , be fixed open rectangles (see Figure 1). Then the domain can be represented as . Let be the boundary of the rectangle and . We define a square grid on , , with step such that the boundary lies entirely on the grid lines. denotes the set of grid nodes on , denotes the set of nodes on and . We refer to the set of nodes on the closure of as , the set of nodes on as and the set of remaining nodes on as . We also introduce the natural number , and , . On the arc , we choose the points ,  , denote the set of these points by and let .

Let , where is the given function in (2). We introduce a gluing operator , ([5] for and [24] for ) at the points of the set . We denote by the approximate solution of the problem (1), (2) obtained by the 5-point scheme with for , and by the 9-point scheme with for , on the “singular” (“nonsingular”) part of . The operator is defined at each point in the following way: we consider the set of all rectangles in the intersections of which the point lies, and we choose one of these rectangles part of whose boundary situated in is furthest away from . The value at the point is computed according to the values of the function at the four vertices , , of the closure of the cell, containing the point of the grid constructed on by multilinear interpolation.

The value of at the point is expressed linearly in terms of the values of at the points , , of the grid constructed on some part of whose boundary located in is the maximum distance away from , and in terms of the boundary values of , at a fixed number of points. Moreover, has the representation where , and for the exact solution of the problem (1), (2), we have

Remark 2. Let be the set of such points , for which all points in the expression (6) are in . If some of the points in (6) emerge through the side when , , we denote the set of such points by . Then, according to the construction of in [4] the expression at each point can be expressed as follows: where corresponds to such point for which the line is perpendicular to .

Consider for each the following system of linear algebraic equations: where

Theorem 3. There is a natural number such that for all , and for each , the system (11)–(14) has a unique solution.

Proof. The proof follows when from [5], and when from [3, 4].

We consider the sector , and let , , be the values of the solution of the system (11)–(14) on (at the quadrature nodes). The function defined on is called an approximate solution of the problem (1), (2) on the closed block .

Everywhere below we will denote constants which are independent of and of the cofactors on their right by for simplicity.

Theorem 4. There exists a natural number such that for where is a fixed number, the following inequalities are valid: for all . Everywhere , is a solution of the problem (1), (2).

Proof. The proof is carried out analogically to the proof of Theorems 1 and 2 in [3].

4. Computational Algorithm

Let , where , , is a fixed number, and and are integers. We introduce a square grid with the lines , , , is an integer, , . Let , and be a set of nodes on (the boundary of ) where

We consider for each the finite difference problem where is a given function on that vanishes at the end points.

The solution of the problem (25) can be found using the superposition principle as the sum of solution of four problems of the type

The solution of the problem (26), when has the representation where for the 5-point approximation [11], for the 9-point approximation [12].

The Discrete Fast Fourier Transform is used for the realization of the finite sums in (27). The solution of the problem (26), for can be represented analogously.

Now we describe the algorithm of implementing the BGM for the slit problem.

Step 1. Suppose that we have zero approximation to the exact solution of (11)–(14).

Step 2. Finding by the formula (13) on we solve the system (11), (12) on each grid by using the representation of finite difference solution described before Step 1.

Step 3. Using (6) we calculate the values at the quadrature nodes for each , by the formula (14).

Step 4. Repeating Steps 2 and 3 we have the sequence , of Schwarz's iterations defined as follows:

As a stopping criteria of the Schwarz's iterations (30), we use the inequality for the prescribed accuracy of .

Step 5. Let , , , in (14) be the values at the quadrature nodes on for the final iteration . Using these values we can calculate the value of the solution at any point in the singular part by the explicit formula

5. Numerical Results

The computational algorithm in Section 4 is applied and the implementation of the block-grid method is carried out using double precision. Let , , be the errors in the “singular” part and be the error in the “nonsingular” part of the domain .

In Table 1 the errors are given by the BGM when 5-point scheme with interpolation is used, and the iterations are terminated by using . Table 2 represents the errors by the BGM when 9-point scheme is used with and the stopping criteria for the Schwarz's iterations is taken as .

tab1
Table 1: The errors by BGM when 5-point scheme is used with interpolation.
tab2
Table 2: The errors by BGM when 9-point scheme is used with interpolation.

The order of convergence in the “nonsingular” part, and the order of convergence in the “singular” part of are respectively, where is a positive integer, is the final iteration number (Section 4), . Taking , , Tables 3 and 4 represent the order of convergence of the BGM in the “nonsingular” part and the “singular” part of the domain for and , respectively.

tab3
Table 3: The order of convergence , and when .
tab4
Table 4: The order of convergence , and when .

The obtained numerical results in Tables 3 and 4 show that the order of convergence of the approximate solution is for the 5-point scheme with interpolation and it is for the 9-point scheme with interpolation in the “nonsingular” part. In both tables, the order of convergence in the “singular” part is higher than the order of convergence in the “nonsingular” part of the domain, which justifies the estimation (19) in Theorem 4. The errors with respect to the number of quadrature nodes in the “singular” part and in the “nonsingular” part by BGM for , and are given in Figures 2 and 3, respectively. These figures demonstrate that the error in the “singular” part is less than the error in the “nonsingular” part for sufficiently large as it follows from the estimation (19) in Theorem 4. The graphical results in Figures 49 are obtained by the BGM when 9-point scheme is used with interpolation for , . In Figure 4, the highly accurate approximate solution and the exact solution is illustrated. Figure 5 represents the decrease of the error function in the “singular” part of the domain as approaches to zero, which agrees with the estimation (19) in Theorem 4. Moreover, on the “singular” part, up to second order derivatives of the solution at grid points are approximated effectively by a simple differentiation of the function (31). The shapes of the first partial derivatives , are demonstrated in Figure 6 and the shapes , are given in Figures 7, 8, and 9, respectively, to show the singular behaviour of the solution around the singular point.

948564.fig.002
Figure 2: The errors with respect to number of quadrature nodes , in the “singular” part and in the “nonsingular” part by the BGM when 5-point scheme is used with for .
948564.fig.003
Figure 3: The errors with respect to number of quadrature nodes , in the “singular” part and in the “nonsingular” part by the BGM when 9-point scheme is used with for .
fig4
Figure 4: The highly accurate approximate solution and the exact solution in the “singular” part for , .
948564.fig.005
Figure 5: The error in the “singular” part for , .
fig6
Figure 6: The first partial derivatives and in the “singular” part for , .
948564.fig.007
Figure 7: The second partial derivative in the “singular” part for , .
948564.fig.008
Figure 8: The second partial derivative in the “singular” part for , .
948564.fig.009
Figure 9: The mixed partial derivative in the “singular” part for , .
5.1. Stress Intensity Factor

In engineering problems a very important constant is the so-called stress intensity factor . This constant gives a measure of “the amount of torsion the beam can endure before fracture occurs” [10, 13]. On the basis of (31) we give a simple and highly accurate formula for the stress intensity factor denoting by for : where and is the final iteration number. The exact value of the stress intensity factor is . For fixed number of quadrature nodes , the second column in Table 5 represents the error of the stress intensity factor when 5-point scheme is used with and the last column represents this error when 9-point scheme is used with .

tab5
Table 5: The error of the stress intensity factor for fixed .

6. Conclusion

For the solution of the Laplace equation on polygons with a slit, the real characteristics of the block-grid method is investigated. The given polygon is decomposed into five overlapping rectangles and one sector. In the sector, we approximate the special integral representation of the solution, which takes into account the behaviour of the exact solution near the end point of the slit. On the rectangles, to approximate Laplace's equation on square grids either 5-point scheme is used which is simpler by means of sparsity, or 9-point scheme is used, which gives a highly accurate approximation. In correspondence with the finite difference scheme used, a gluing together of the subsystems is carried out effectively by a sufficiently simple linear interpolation , or a highly accurate interpolation . By choosing the step size , the obtained numerical results show that the order of convergence of the approximate solution is for the 5-point scheme with and it is for the 9-point scheme with in the “nonsingular” part. The results also show that the order of convergence in the “singular” part is higher than the order of convergence in the “nonsingular” part of the domain. This conclusion justifies the theoretical results obtained in [25]. Moreover, the shapes up to the second-order derivatives of the highly accurate solution obtained by the BGM are shown to display the singular behaviour at the end point of the slit. Finally the stress intensity factor is approximated by the given highly accurate formula.

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