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

In this paper, we derive the convergence for the high-accuracy algorithm in solving the Dirichlet problem of the modied Helmholtz equation. By the boundary element method, we transform the system to be a boundary integral equation. e highaccuracy algorithm using the specic quadrature rule is developed to deal with weakly singular integrals. e 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 O(h0). e numerical examples support the theoretical analysis.


Introduction
Modi ed Helmholtz equation arises in many important elds 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][6][7][8][9][10][11] can be used to solve the modi ed 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 de ned on the boundary. In the literature, several popular numerical methods for solving the boundary integral equations reformulated from the modi ed 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 modi ed Helmholtz equation with the mixed boundary value problem; Kropinski and Quaife [13] used the fast multipole-accelerated integral equation methods to solving the modi ed Helmholtz equation with linear boundary conditions, and so on. Motivation of this work is to present an accuracy algorithm for solving the following modi ed Helmholtz problem: where a is a real and positive constant and called the acoustic wave number. Ω ⊂ R 2 is a bounded domain with the boundary Γ ∪ d m 1 Γ m , d > 1 which is a closed curve. g(x) is a given function.
Based on the boundary element method, the function u(y) satis es where x (x 1 , x 2 ), y (y 1 , y 2 ), and v(x) is the density function, and where c is Euler's constant.
Since u(y) is a continuous function in Ω ∪ Γ, we have g(y) � Γ K * (y, x)v(x)ds x , y ∈ Γ. (5) e aforementioned equation is weakly singular boundary integral equation of first kind, whose solution exists and is unique as long as C T ≠ 1 [15], where C T is the logarithmic capacity. As soon as v(x) is solved from (5), the function u(y)(y ∈ Ω) 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. e 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.
is 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.

Singularity of Integral Kernels and Solutions
Let Γ � ∪ d m�1 Γ m , (d > 1) be closed polygons with C Γ≠1 and Γ m (m � 1, . . . , d) be a piecewise smooth curve. Define the boundary integral operators on Γ m : where k qm (y, x) � K * (y, x). en, (5) can be converted to a matrix operator equation: where ) T , and G � (g 1 (y), g 2 (y), . . . , g d (y)) T . Assume that Γ m be described by the parameter mapping: x m (s) � (x m 1 (s), x m 2 (s)): Using the sin p -transformation [17], where ϑ p (τ) � τ 0 (sin(πρ)) p dρ. e operators in (6) are converted to the following integral operators on [0,1]: where k qm (t, τ) Hence, (7) is equivalent to where ω � (ω 1 , ω 2 , . . . , ω d ) T . 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 en, (10) becomes where Since ψ p (t) ∈ C ∞ [0, 1] increases monotonously on [0,1] with ψ p (0) � 0 and ψ p (1) � 1, the solutions of (14) are equivalent to those of (7) [17]. Now, let us study the solution singularity for (7). Denote the corner points Q 1 , . . . , Q d of the closed polygonal boundary Γ and for the middle corner Q m . Based on the potential theory [15,18] Although v m (s) has a singularity, ω m (s) has no singularity by (8). e kernel b mm of B mm has a singularity at corners, but sin 2 (πs)b mm has no singularity.

The High-Accuracy Algorithm
Lemma 1 (see [19]). Assume that g(x) and ere is the following quadrature formula: (16) and the error where h is the mesh width and ζ ′ (t) is the derivative of the Riemann zeta function.
Obviously, (25) is a system of linear equations with n(� d m�1 n m ) unknowns. Once ω h is solved by (25), the solution u(y)(y ∈ Ω) can be computed by

Convergence of the Algorithm
Form (11) and (26), we have Lemma 2 (see [21]). Based on Lemma 2, we immediately get the following corollary:

Corollary 1. A h is invertible, and (A h ) − 1 is uniformly bounded.
From Corollary 1, we know (25) is equivalent to where E h denotes the unit matrix. Now we give the following definitions to discuss the approximate convergence in (29). Define be a piecewise linear function subspace with base points t i n m i�1 and e j (t), j � 1, . . . , n m be base function satisfying e j (t i ) � δ ji . Define a prolongation operator P h m : G � P h G h , we construct an operator equation where P h � diag(P 1 , . . . , P d ), R h � diag(R 1 , . . . , R d ). Obviously, if ω h is the solution of (32), then R h ω h must be the solution of (29); conversely, if ω h is the solution of (29), then P h ω h must be the solution of (32).
In order to prove the convergence of the algorithm, we give the following lemma.
where the kernel of B qm is ψ p ′ b qm .

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

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 O(h 3 0 ). ese 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
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.