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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 864865, 11 pages
http://dx.doi.org/10.1155/2013/864865
Research Article

A Fourth-Order Block-Grid Method for Solving Laplace's Equation on a Staircase Polygon with Boundary Functions in

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

Received 30 April 2013; Accepted 11 May 2013

Academic Editor: Allaberen Ashyralyev

Copyright © 2013 A. A. Dosiyev and 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 integral representations of the solution around the vertices of the interior reentered angles (on the “singular” parts) are approximated by the composite midpoint rule when the boundary functions are from These approximations are connected with the 9-point approximation of Laplace's equation on each rectangular grid on the “nonsingular” part of the polygon by the fourth-order gluing operator. It is proved that the uniform error is of order where and is the mesh step. For the -order derivatives () of the difference between the approximate and the exact solutions, in each “ singular” part order is obtained; here is the distance from the current point to the vertex in question and is the value of the interior angle of the th vertex. Numerical results are given in the last section to support the theoretical results.

1. Introduction

In the last two decades, among different approaches to solve the elliptic boundary value problems with singularities, 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, 2], and references therein).

In [27], a new combined difference-analytical method, called the block-grid method (BGM), is proposed for the solution of the Laplace equation on polygons, when the boundary functions on the sides causing the singular vertices are given as algebraic polynomials of the arc length. In the BGM, the given polygon is covered by a finite number of overlapping sectors around the singular vertices (“singular” parts) and rectangles for the part of the polygon which lies at a positive distance from these vertices (“nonsingular” part). The special integral representation in each “singular” part is approximated, and they are connected by the appropriate order gluing operator with the finite difference equations used in the “nonsingular” part of the polygon.

In [8, 9], the restriction on the boundary functions to be algebraic polynomials on the sides of the polygon causing the singular vertices in the BGM was removed. It was assumed that the boundary function on each side of the polygon is given from the Hölder classes , and on the “nonsingular” part the 5-point scheme is used when [8] and the 9-point scheme is used when [9]. For the 5-point scheme a simple linear interpolation with 4 points is used. For the 9-point scheme an interpolation with 31 points is used to construct a gluing operator connecting the subsystems. Moreover, to connect the quadrature nodes which are at a distance of less than from boundary of the polygon, a special representation of the harmonic function through the integrals of Poisson type for a half plane is used (see [9]).

In this paper the BGM is developed for the Dirichlet problem when the boundary function on each side of the polygon is from , by using the 9-point scheme on the “nonsingular” part with 16-point gluing operator for all quadrature nodes, including those near the boundary. The paper is organized as follows: in Section 2, the boundary value problem and the integral representations of the exact solution in each “singular” part are given. In Section 3, to support the aim of the paper, a Dirichlet problem on the rectangle for the known exact solution from ,  , is solved using the 9-point scheme and the numerical results are illustrated. In Section 4, the system of block-grid equations and the convergence theorems are given. In Section 5 a highly accurate approximation of the coefficient of the leading singular term of the exact solution (stress intensity factor) is given. In Section 6 the method is illustrated for solving the problem in L-shaped polygon with the corner singularity. The conclusions are summarized in Section 7.

2. Dirichlet Problem on a Staircase Polygon

Let be an open simply connected polygon, , , its sides, including the ends, enumerated counterclockwise, the boundary of , and , , the interior angle formed by the sides and . Denote by the vertex of the th angle and by a polar system of coordinates with a pole in , where the angle is taken counterclockwise from the side .

We consider the boundary value problem

where is a given function on of the arc length taken along , and ; that is, has the fourth-order derivative on , which satisfies a Hölder condition with exponent .

At some vertices for the conjugation conditions

are fulfilled. For the remaining vertices , the values of and at might be different. Let be the set of all for which or , but (2) is not fulfilled. In the neighborhood of , we construct two fixed block sectors , where .

Let (see [10]) where

The function is harmonic on and satisfies the boundary conditions in (1) on and , , except for the point when.

We formally set the value of and the solution of the problem (1) at the vertex equal to .

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

Lemma 1 (see [10]). The solution of the boundary value problem (1) can be represented on , in the form
where is the curvilinear part of the boundary of  , and is the function defined by (4).

3. 9-Point Solution on Rectangles

Let be a rectangle, with being rational, the sides, including the ends, enumerated counterclockwise, starting from the left side , and the boundary of .

We consider the boundary value problem

where is the given function on .

Definition 2. One says that the solution of the problem (9) belongs to if and at the vertices the conjugation conditions are satisfied.

Remark 3. From Theorem 3.1 in [11] it follows that the class of functions is wider than .

Let , with , integers. We assign to a square net on , with step , obtained with the lines ,. Let be a set of nodes on the interior of and let

We consider the system of finite difference equations where

On the basis of the maximum principle the unique solvability of the system of finite difference equations (13) follows (see [12, Chapter 4]).

Everywhere below we will denote constants which are independent of and of the cofactors on their right by , generally using the same notation for different constants for simplicity.

Theorem 4. Let be the solution of problem (9). If , then
where is the solution of the system (13).

Proof. For the proof of this theorem see [13].

Let and let be the boundary of . We consider the Dirichlet problem

where , is the exact solution of this problem, which is from .

We solve the problem (16) by approximating 9-point scheme when for the different values of .

In Figure 1, the order of numerical convergence of the 9-point solution , for different and , is demonstrated. These results show that the order of numerical convergence, when the exact solution , depends on and and is when , which supports estimation (15). Moreover, this dependence also requires the use of fourth-order gluing operator for all quadrature nodes in the construction of the system of block-grid equations, when the given boundary functions are from the Hölder classes .

864865.fig.001
Figure 1: Dependence of the approximate solutions for the boundary functions from .

4. System of Block-Grid Equations

In addition to the sectors and (see Section 2) in the neighborhood of each vertex of the polygon , we construct two more sectors and , where , and , , and let .

We cover the given solution domain (a staircase polygon) by the finite number of sectors , and rectangles ,  , as is shown in Figure 2, for the case of -shaped polygon, where (see also [2]). It is assumed that for the sides and of the quotient is rational and . Let be the boundary of the rectangle , let be the curvilinear part of the boundary of the sector , and let . We choose a natural number and define the quantities , and . On the arc we take the points , and denote the set of these points by . We introduce the parameter , where is a gluing depth of the rectangles , and define a square grid on , with maximal possible step such that the boundary lies entirely on the grid lines. Let be the set of grid nodes on , let be the set of nodes on , and let . We denote the set of nodes on the closure of by , the set of nodes on by , and the set of remaining nodes on by .

864865.fig.002
Figure 2: Description of BGM for the L-shaped domain.

Let

Let , where , is the given function in (1). We use the matching operator at the points of the set constructed in [14]. 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 ,

Let be the set of such points , for which all points in expression (19) are in . If some points in (19) emerge through the side , then the set of such points is denoted by . According to the construction of in [14], the expression at each point can be expressed as follows: where

and corresponds to such a point for which the line is perpendicular to .

Let The quantities in (24) are given by (4) and (5), which contain integrals that have not been computed exactly in the general case. Assume that approximate values and of the quantities in (24) are known with accuracy ; that is,

where , , and , are constants independent of .

Consider the system of linear algebraic equations

where , and  .

Let be the sector, where , and let ,  ,  , be the solution values of the system (26) on (at the quadrature nodes). The function defined on , is called an approximate solution of the problem (1) on the closed block ,  .

Definition 5. The system (26) and (27) is called the system of block-grid equations.

Theorem 6. There is a natural number , such that for all and for any the system (26) has a unique solution.

Proof. From the estimation in [15] follows the existence of the positive constants and , such that for all The proof is obtained on the basis of principle of maximum by taking into account (14), (19), (20), and (28).

Theorem 7. There exists a natural number , such that for all where is a fixed number, and for any the following inequalities are valid: for integer when , for any , if , for noninteger , when . Everywhere is the exact solution of the problem (1) and is defined by formula (27).

Proof. Let where is a solution of system (26) and is the trace on of the solution of (1). On the basis of (1), (26), and (34) the error satisfies the system of difference equations where ,  ,
On the basis of estimations (15), (21), (25), and Lemma 1 by analogy to the proof of Theorem 4.3 in [9] the proof of inequality (30) follows.
The function given by formula (27), defined on the closed sector , where , and the integral representation (8) of the exact solution of the problem (1) is given on ,  , and then the difference function is defined on ,   and On the basis of Lemma 6.11 from [16], (25), and (28), for , is a fixed number, and we obtain Furthermore, the function continuous on is a solution of the following Dirichlet problem: Since , on the basis of (39) and (40), from Lemma 6.12 in [16], inequalities (31)–(33) of Theorem 7 follow.

5. Stress Intensity Factor

Let, in the condition , the exponent be such that

where is the symbol of fractional part. These conditions for the given can be fulfilled by decreasing .

On the basis of Section 2 of [11], a solution of the problem (1) can be represented in , as follows: where ,   is the integer part, ,   and are some numbers, and is the harmonic on . By taking , from the formula (43), it follows that the coefficient which is called the stress intensity factor can be represented as

From formula (44) it follows that can be approximated by

Using formula (3), (4), and (27) from (45) for the stress intensity factor (see [17]), we obtain the next formula: This formula is obtained for the second-order BGM in [8].

6. Numerical Results

Let be L-shaped and defined as follows:

where and is the boundary of .

We consider the following problem: where

is the exact solution of this problem.

We choose a “singular” part of as

where . Then is a “nonsingular” part of .

The given domain is covered by four overlapping rectangles , , and by the block sector (see Figure 2). For the boundary of on is the polygonal line . The radius of sector is taken as . According to (49), the function in (4) is

where and  . Since we have only one singular point, we omit subindices in (51). We calculate the values and on the grids , with an accuracy of using the quadrature formulae proposed in [10].

On the basis of (46) and (51), for the stress intensity factor, we have

Taking the zero approximation , the results of realization of the Schwarz iteration (see [2]) for the solution of the problem (48) are given in Tables 1, 2, 3, and 4. Tables 1 and 2 represent the order of convergence. Table 6 shows a highly accurate approximation of the stress intensity factor by the proposed fourth order BGM in the “nonsingular” and the order of convergence

tab1
Table 1: The order of convergence in the “nonsingular” part when and .
tab2
Table 2: The order of convergence in the “singular” part when and .
tab3
Table 3: The minimum errors of the solution over the pairs in maximum norm when .
tab4
Table 4: In , the minimum errors of the derivatives over the pairs in maximum norm when .

in the “singular” parts of , respectively, for , where is a positive integer. In Table 3, the minimal values over the pairs of the errors in maximum norm, of the approximate solution when , are presented. The similar values of errors for the first-order derivatives are presented in Table 4, when and are approximated by fourth-order central difference formula on for . For , the order of errors decreases down to , which are presented in Table 5. This happens because the integrands in (51) are not sufficiently smooth for fourth-order differentiation formula. The order of accuracy of the derivatives for can be increased if we use similar quadrature rules, which we used for the integrals in (51) for the derivatives of integrands also.

tab5
Table 5: In , the minimum errors of the derivatives over the pairs in maximum norm when .
tab6
Table 6: The stress intensity factor for when .

Figures 3 and 4 show the dependence on for different mesh steps . Figure 5 demonstrates the convergence of the BGM with respect to the number of quadrature nodes for different mesh steps . The approximate solution and the exact solution in the “singular” part are given in Figure 6, to illustrate the accuracy of the BGM. The error of the block-grid solution, when the function in (51) is calculated with an accuracy of , is presented in Figure 7. Figures 8 and 9 show the singular behaviour of the first-order partial derivatives in the “singular” part. The ratios and , when with respect to different values for and for a fixed value of of , are illustrated in Figures 10 and 11, respectively. These ratios show that the order of the convergence in both the “singular” and the “nonsingular” parts is asymptotically equal to when is kept fixed for , and it is selected as large as possible for .

fig3
Figure 3: Dependence on for .
fig4
Figure 4: Dependence on for .
fig5
Figure 5: Maximum error depending on the number of quadrature nodes .
fig6
Figure 6: The approximate solution and the exact solution in the “singular” part for .
864865.fig.007
Figure 7: The error function in “singular” part when .
864865.fig.008
Figure 8: in the “singular” part.
864865.fig.009
Figure 9: in the “singular” part.
864865.fig.0010
Figure 10: when by fixing for for different values of .
864865.fig.0011
Figure 11: when by fixing for for different values of .

7. Conclusions

In the block-grid method (BGM) for solving Laplace's equation, the restriction on the boundary functions to be algebraic polynomials on the sides of the polygon causing the singular vertices is removed. This condition is replaced by the functions from the Hölder classes ,  . In the integral representations around singular vertices (on the “singular” part), which are combined with the 9-point finite difference equations on the “nonsingular” part of the polygon, the boundary conditions are taken into account with the help of integrals of Poisson type for a half-plane. To connect the subsystems, a homogeneous fourth-order gluing operator is used. It is proved that the final uniform error is of order , where is the error of the approximation of the mentioned integrals and is the mesh step. For the -order derivatives () of the difference between the approximate and the exact solutions, in each “singular” part order is obtained. The method is illustrated in solving the problem in L-shaped polygon with the corner singularity. Dependence of the approximate solution and its errors on and the number of quadrature nodes are demonstrated. Furthermore, by the constructed approximate solution on the “singular” part of the polygon, a highly accurate formula for the stress intensity factor is given.

From the error estimation formula (33) of Theorem 7 it follows that the error of the approximate solution on the block sectors decreases as , which gives an additional accuracy of the BGM near the singular points.

The method and results of this paper are valid for multiply connected polygons.

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