Complexity
Volume 2020 (2020), Article ID 6484890, 8 pages
https://doi.org/10.1155/2020/6484890

Research Article

Convergence of the High-Accuracy Algorithm for Solving the Dirichlet Problem of the Modified Helmholtz Equation

Hu Li¹ and Guang Zeng²

¹School of Mathematics, Chengdu Normal University, Chengdu 611130, China
²School of Sciences, East China University of Technology, Nanchang 330013, China

Correspondence should be addressed to Hu Li; lihu_0826@163.com

Received 19 September 2019; Accepted 26 December 2019; Published 20 January 2020

Academic Editor: Toshikazu Kuniya

Copyright © 2020 Hu Li and Guang Zeng. 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.

In this paper, we derive the convergence for the high-accuracy algorithm in solving the Dirichlet problem of the modified Helmholtz equation. By the boundary element method, we transform the system to be a boundary integral equation. The high-accuracy algorithm using the specific quadrature rule is developed to deal with weakly singular integrals. The convergence of the algorithm is proved based on Anselone’s collective compact theory. Moreover, an asymptotic error expansion shows that the algorithm is of order . The numerical examples support the theoretical analysis.

1. Introduction

Modified Helmholtz equation arises in many important fields of science and engineering, for instance, wave propagation and scattering [1], structural vibration [2], implicit marching schemes for the heat equation [3], the Navier–Stokes equations [4], and so on. With the rapid development of computer power, many numerical methods [5–11] can be used to solve the modified Helmholtz equation. Among all methods for numerically solving the boundary value problem, the boundary integral equation approach is one of the most fundamental treatments. By the fundamental solutions, it can transform the boundary problem into an integral equation defined on the boundary. In the literature, several popular numerical methods for solving the boundary integral equations reformulated from the modified Helmholtz equation are used. For instance, Steinbach and Tchoualag [12] used the Galerkin method with one-periodic B-spline as basis functions and a spectral collocation method to solving the modified Helmholtz equation with the mixed boundary value problem; Kropinski and Quaife [13] used the fast multipole-accelerated integral equation methods to solving the modified Helmholtz equation with linear boundary conditions, and so on. Motivation of this work is to present an accuracy algorithm for solving the following modified Helmholtz problem:

where a is a real and positive constant and called the acoustic wave number. is a bounded domain with the boundary which is a closed curve. is a given function.

Based on the boundary element method, the function satisfies

where , , and is the density function, and

is the fundamental solution of the modified Helmholtz equation. is the modified Bessel function [14]:

where γ is Euler’s constant.

Since is a continuous function in , we have

The aforementioned equation is weakly singular boundary integral equation of first kind, whose solution exists and is unique as long as [15], where is the logarithmic capacity. As soon as is solved from (5), the function can be calculated by (2). Galerkin methods and collocation methods have been used to solve (5) based on the projective theory. However, there exist the following disadvantages: (a) the discrete matrix is full and each element has to calculate the weakly singular integral for collocation methods or the double weakly singular integral for Galerkin methods; (b) the order of accuracy is lower [16].

In this paper, we present a high-accuracy algorithm to solve (5). By specific quadrature rule, the high-accuracy algorithm is constructed to discrete the boundary integral equation, and a linear system is obtained. The calculation of the discrete matrix becomes very simple and straightforward without any singular integrals. We prove the convergence of the linear system by estimating eigenvalues of the discrete matrix and Anselone’s collective compact theory. Moreover, an asymptotic expansion of errors is obtained. Finally, numerical results verify the theoretical analysis.

This paper is organized as follows: in Section 2, the singularity of integral kernels and solutions are discussed; in Section 3, the high-accuracy algorithm is shown; in Section 4, the convergence analysis of the algorithm is given; in Section 5, an asymptotic expansion of the errors is obtained; and in Section 6, numerical examples are shown.

2. Singularity of Integral Kernels and Solutions

Let be closed polygons with and be a piecewise smooth curve. Define the boundary integral operators on :

where .

Then, (5) can be converted to a matrix operator equation:

where , , and . Assume that be described by the parameter mapping: with , . Using the -transformation [17],

where . The operators in (6) are converted to the following integral operators on [0,1]:

where , , , and .

Hence, (7) is equivalent to

where . By (4), it is clear that the operator K is a weakly singular integral operator, which is decomposed into the sum of the integral operators A and B, where one carries the main singularity characteristic of A and the other is a compact operator with a smooth kernel.

Define

and

where and

where    as .

Then, (10) becomes

where , , and . is an isometry operator from to for any real number r. A is also an isometry operator from to . Hence, A is invertible, and (14) is equivalent to

Since increases monotonously on [0,1] with and , the solutions of (14) are equivalent to those of (7) [17].

Now, let us study the solution singularity for (7). Denote the corner points of the closed polygonal boundary and . Denote by for the endpoints and or for the middle corner . Based on the potential theory [15, 18], near the corner or endpoint , the solution has a singularity , where , and at is the arc parameter. Since the singularity of results from the singularities of the potential , when is nonsingular in the exterior region, the singularity of becomes , and when it is nonsingular in the interior, the singularity of becomes . Although has a singularity, has no singularity by (8). The kernel of has a singularity at corners, but has no singularity.

3. The High-Accuracy Algorithm

Lemma 1 (see [19]). Assume that and . is a periodic function with a period of and at least 2 m differentiable in . There is the following quadrature formula:

and the error

where h is the mesh width and is the derivative of the Riemann zeta function.

Let be mesh widths and be nodes.

(1)Since are the smooth functions on [0, 1] for , by the trapezoidal or the midpoint rule [20], we construct the Nyströms approximate operator of the integral operator defined by

 We have the following error bounds [20]:

(a)For ,

(b)For ,

 where [17]

(2)Since have the logarithmic singularities on [0,1], by Lemma 1, we get the following approximations of the integral operator defined by

and the error bounds

The approximations of the integral operator are defined by

Setting , we obtain the discrete equations of (14):

where , , , , , and .

Obviously, (25) is a system of linear equations with unknowns. Once is solved by (25), the solution can be computed by

4. Convergence of the Algorithm

Form (11) and (26), we have

Lemma 2 (see [21]). (1) There exists a positive so that the eigenvalues of satisfy ; (2) is invertible, and is uniformly bounded.

Based on Lemma 2, we immediately get the following corollary:

Corollary 1. is invertible, and is uniformly bounded.

From Corollary 1, we know (25) is equivalent to

where denotes the unit matrix.

Now we give the following definitions to discuss the approximate convergence in (29).

Define a subspace of the space with a norm . Let be a piecewise linear function subspace with base points and be base function satisfying . Define a prolongation operator satisfying

Define a restricted operator satisfying

Replacing , , and , we construct an operator equation

where , . Obviously, if is the solution of (32), then must be the solution of (29); conversely, if is the solution of (29), then must be the solution of (32).

In order to prove the convergence of the algorithm, we give the following lemma.

Lemma 3 (see [21]). The operator sequence is uniformly bounded and convergence to the embedding operator I.

Lemma 4 (see [22]). Let the integral operator and exist, and let the integral operator with the kernel of satisfy , where is a linear bound operator space, and is the kernel of . Also assume that and are continuous on . Let be nodes. Then, the Nyström approximation of ,

is collectively compact convergent to ,

Based on Lemma 4, we can immediately obtain the following corollary.

Corollary 2. Let with , and let or . Then, under (8), the Nyström approximation of has

by the trapezoidal or the midpoint rule, where denotes the collectively compact convergence.

Lemma 5. Let with , and let or . Then, we have

where the kernel of is .

Proof. the kernel of has logarithmic singularity at the corner with an inner angle (replacing when ). Without loss of generality, suppose that , then by (4), the kernel of can be expressed by

where and . We assume that , namely, has the logarithmic singularity at points and are continuously differentiable in . Consider

We have

So, the singularity of occurs in . In order to remove the singularity, define

Since have p order zero, we have the error estimate

which shows that is bounded and continuous on when for any .
Similarly, we can prove and are continuous on , which can be obtained by

and . As above, is also continuous on . Thus, and are continuous on . By Lemma 4, we complete Lemma 5.

Theorem 1. Let with and be the smooth curve. The operator sequence is collectively convergent to in ,

Proof. Let be a unit ball. is the grid step sequence, where with , . We arbitrarily take a sequence in space , where with , . We firstly consider the first component of :

For or , by Corollary 2, in , and there exists a convergent subsequence in . For , then

By Lemma 5, there exists a convergent subsequence in . Based on the above two cases, it is proved that there exists an infinite subsequence such that the first component converge. Similarly, it can be concluded that there exists an infinite subsequence such that converge. Therefore, is collectively compact convergent sequence, and , where shows the point convergence. We complete the proof.

5. The Error of the Algorithm

In this section, we derive the asymptotic expansion of the errors. We first provide the main result.

Theorem 2. Let with . There exists a vector function independent of such that the following multiparameter asymptotic expansion holds at nodes

where , .

Proof. By (19), (20), and (23), we can obtain

where , . Since , we have

Constructing the auxiliary equation

and its approximate equation

and substituting (50) in (49), we obtain

Since is uniformly bounded from Theorem 1, we get

Replacing with , we can complete the proof.

6. Numerical Examples

In this section, in order to verify the errors in the previous sections, we carry out some numerical examples for the modified Helmholtz equation with Dirichlet boundary value problems by the high-accuracy algorithm.

Example 1. We consider the modified Helmholtz equation with and on a plate domain with the boundary and and . We consider the Dirichlet boundary conditions corresponding to the exact solution .
We calculate the errors of at the points (0.5, 0.3) and (0.5, 0.35) by . The errors are given in Table 1. Average errors of 100 points are given in Table 2. We let ratio denote . From the numerical results, we can see that , which shows that the accuracy order of is for the high-accuracy algorithm.

Table 1: The simulation results for Example 1.

(8, 8)
1.3001e − 3
—
1.3000e − 3
—
1.5002e − 3
—
1.5000e − 3
—

(16, 16)
1.5737e − 4
8.2162
1.6175e − 4
8.0786
1.7826e − 4
8.5450
1.6922e − 4
8.8212

(32, 32)
1.9508e − 5
8.0668
2.0317e − 5
7.9613
2.2899e − 5
7.7844
2.1061e − 5
8.0350

(64, 64)
2.4256e − 6
8.0428
2.5402e − 6
7.9983
2.8596e − 6
8.0078
2.6396e − 6
7.9789

(128, 128)
2.9535e − 7
8.2125
3.1267e − 7
8.1243
3.5122e − 7
8.1420
3.2650e − 7
8.0844

(256, 256)
3.2949e − 8
8.9638
3.6475e − 8
8.5720
4.0697e − 8
8.6301
3.8832e − 8
8.4081

Table 2: The simulation results for Example 1.

(8, 8)
1.4512e − 4
—
1.4195e − 4
—

(16, 16)
1.7060e − 5
8.5065
1.6596e − 5
8.5534

(32, 32)
2.1358e − 6
7.9874
2.0716e − 6
8.0113

(64, 64)
2.6607e − 7
8.0271
2.5935e − 7
7.9876

(128, 128)
3.2564e − 8
8.1709
3.2011e − 8
8.1019

(256, 256)
3.7158e − 9
8.7636
3.7756e − 9
8.4786

Example 2. We consider the modified Helmholtz equation with and on a plate domain with the boundary and and . We consider the Dirichlet boundary conditions corresponding to the exact solution .
We calculate the errors of at the points (0.6, 0.6) and (0.6, 0.5) by . The errors are given in Table 3. Average errors of 100 points are given in Table 4. We let ratio denote . From the numerical results, we can see that , which shows that the accuracy order of is for the high-accuracy algorithm.

Table 3: The simulation results for Example 2.

(8, 8)
4.2759e − 4
—
5.0447e − 4
—
5.7375e − 4
—
7.8778e − 4
—

(16, 16)
4.2257e − 5
10.1188
5.9981e − 5
8.4105
6.7300e − 5
8.5252
9.6859e − 5
8.1332

(32, 32)
5.2327e − 6
8.0756
7.5075e − 6
7.9895
8.3529e − 6
8.0571
1.2148e − 5
7.9733

(64, 64)
6.3995e − 7
8.1768
9.3323e − 7
8.0446
1.0259e − 6
8.1423
1.5119e − 6
8.0349

(128, 128)
7.3252e − 8
8.7362
1.1356e − 7
8.2182
1.1946e − 7
8.5879
1.8458e − 7
8.1913

(256, 256)
5.7950e − 9
12.6406
1.2628e − 8
8.9928
1.0553e − 8
11.3198
2.0826e − 8
8.8628

Table 4: The simulation results for Example 2.

(8, 8)
4.9001e − 4
—
6.3035e − 4
—

(16, 16)
5.3408e − 5
9.1748
7.6396e − 5
8.2511

(32, 32)
6.6211e − 6
8.0664
9.5714e − 6
7.9817

(64, 64)
8.1158e − 7
8.1583
1.1905e − 6
8.0396

(128, 128)
9.3748e − 8
8.6571
1.4511e − 7
8.2045

(256, 256)
7.8774e − 9
11.9008
1.6256e − 8
8.9266

7. Conclusions

In the paper, we introduce a high-accuracy algorithm using the specific quadrature rule to solve the modified Helmholtz equation with Dirichlet boundary value problems. A few concluding remarks can be made: (1) evaluation on entries of discrete matrices is very simple and straightforward without any singular integrals by the algorithm. Hence, the algorithm is appropriate to solve singularity problems; (2) the numerical results show that the algorithm retains the optimal convergence order . These are remarkable advantages of the present algorithm, and other existing numerical methods, such as Galerkin methods and collocation methods, did not possess; and (3) this algorithm will be used to solve the three-dimensional axisymmetric boundary integral equations in the future.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This study was supported by the National Natural Science Foundation of China (11661005; 11301070) and the Program of Chengdu Normal University (YJRC2018-1; CS18ZDZ02).

References

L. Marin, L. Elliott, P. J. Heggs, D. B. Ingham, D. Lesnic, and X. Wen, “Conjugate gradient-boundary element solution to the Cauchy problem for Helmholtz-type equations,” Computational Mechanics, vol. 31, no. 3, pp. 367–377, 2003.

D. E. Beskos, “Boundary element methods in dynamic analysis: part II (1986–1996),” Applied Mechanics Reviews, vol. 50, no. 3, pp. 149–197, 1997.

H. Cheng, J. Huang, and T. J. Leiterman, “An adaptive fast solver for the modified Helmholtz equation in two dimensions,” Journal of Computational Physics, vol. 211, no. 2, pp. 616–637, 2006.

L. Greengard and M. C. Kropinski, “An integral equation approach to the incompressible Navier–Stokes equations in two dimensions,” SIAM Journal on Scientific Computing, vol. 20, no. 1, pp. 318–336, 1998.

Y. Gu, C.-M. Fan, and R.-P. Xu, “Localized method of fundamental solutions for large-scale modeling of two-dimensional elasticity problems,” Applied Mathematics Letters, vol. 93, pp. 8–14, 2019.

A. Zhang, Y. Gu, Q. S. Hua, W. Chen, and C. Z. Zhang, “A regularized singular boundary method for inverse Cauchy problem in three-dimensional elastostatics,” Advances in Applied Mathematics and Mechanics, vol. 10, no. 6, pp. 1459–1477, 2018.

F. Wang, C.-M. Fan, Q. Hua, and Y. Gu, “Localized MFS for the inverse Cauchy problems of two-dimensional Laplace and biharmonic equations,” Applied Mathematics and Computation, vol. 364, Article ID 124658, 2020.

F. Wang, Y. Gu, W. Qu, and C. Zhang, “Localized boundary knot method and its application to large-scale acoustic problems,” Computer Methods in Applied Mechanics and Engineering, Article ID 112729, 2019.

F. Wang, W. Chen, W. Qu, and Y. Gu, “A BEM formulation in conjunction with parametric equation approach for three-dimensional Cauchy problems of steady heat conduction,” Engineering Analysis with Boundary Elements, vol. 63, pp. 1–14, 2016.

W. Qu, W. Chen, and Y. Gu, “Fast multipole accelerated singular boundary method for the 3D Helmholtz equation in low frequency regime,” Computers & Mathematics with Applications, vol. 70, no. 4, pp. 679–690, 2015.

W. Qu, “A high accuracy method for long-time evolution of acoustic wave equation,” Applied Mathematics Letters, vol. 98, pp. 135–141, 2019.

O. Steinbach and L. Tchoualag, “Fast fourier transform for efficient evaluation of Newton potential in BEM,” Applied Numerical Mathematics, vol. 81, pp. 1–14, 2014.

M. C. A. Kropinski and B. D. Quaife, “Fast integral equation methods for the modified Helmholtz equation,” Journal of Computational Physics, vol. 230, no. 2, pp. 425–434, 2011.

R. Wang and X. Chen, “A fast solver for boundary integral equations of the modified Helmholtz equation,” Journal of Scientific Computing, vol. 65, no. 2, pp. 553–575, 2015.

Y. Yi and I. Sloan, “On integral equations of the first kind with logarithmic kernel,” Journal of Integral Equations and Applications, vol. 1, no. 4, pp. 549–579, 1988.

Y. Yi, “The collocation method for first-kind boundary integral equations on polygonal regions,” Mathematics of Computation, vol. 54, no. 189, pp. 39–154, 1990.

A. Sidi, “A new variable transformation for numerical integration,” in Integration IV, pp. 359–374, Birkhaurer Verlag, Frankfurt, Germany, 1993.

I. H. Sloan and A. Spence, “The galerkin method for integral equations of the first kind with logarithmic kernel,” IMA Journal of Numerical Analysis, vol. 8, no. 1, pp. 105–122, 1988.

A. Sidi and M. Israeli, “Quadrature methods for periodic singular and weakly singular Fredholm integral equations,” Journal of Scientific Computing, vol. 3, no. 2, pp. 201–231, 1988.

P. Davis, Methods of Numerical Integration, Academic Press, New York, NY, USA, Second edition, 1984.

J. Huang and T. Lu, “Splitting extrapolations for solving boundary integral equations of linear elasticity dirichlet problems on polygons by mechanical quadrature methods,” Journal of Computational Mathematics, vol. 24, no. 1, pp. 9–18, 2006.

X. Luo, J. Huang, and C.-L. Wang, “Splitting extrapolation algorithms for solving the boundary integral equations of anisotropic Darcy’s equation on polygons by mechanical quadrature methods,” Numerical Algorithms, vol. 62, no. 1, pp. 27–43, 2013.