Mechanical Quadrature Method and Splitting Extrapolation for Solving Dirichlet Boundary Integral Equation of Helmholtz Equation on Polygons
We study the numerical solution of Helmholtz equation with Dirichlet boundary condition. Based on the potential theory, the problem can be converted into a boundary integral equation. We propose the mechanical quadrature method (MQM) using specific quadrature rule to deal with weakly singular integrals. Denote by the mesh width of a curved edge of polygons. Then, the multivariate asymptotic error expansion of MQM accompanied with for all mesh widths is obtained. Hence, once discrete equations with coarse meshes are solved in parallel, the higher accuracy order of numerical approximations can be at least by splitting extrapolation algorithm (SEA). A numerical example is provided to support our theoretical analysis.
We consider Helmholtz equation with Dirichlet boundary condition: where is a polygonal domain with the curved boundary , , and the angels between and are in , and the function is known on .
By the potential theory, the solutions of (1) can be represented as a single-layer potential where and is the foundation solution of Helmholtz equation, which is given by And is the Hankel functions of order zero and of the first kind, where for the Bessel function of order zero and for the Neumann function of order zero. And is Euler constant.
In what follows, in order to analyze properties of the kernel, we decompose the kernel where is logarithmic singular function and is a smooth function. is the solution of the following equation: Equation (7) is weakly singular BIE system of the first kind, whose solution exists and is unique as , where is the logarithmic capacity (i.e., the transfinite diameter). Once is solved from (7), the function can be calculated by (2).
Galerkin and collocation methods [2, 3] are used to solve (7). However, 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, which implies that the work calculating discrete matrix is so large as greatly to exceed to solve the discrete equations. When the numerical methods are applied, the accuracy of numerical solutions is lower at singular points  and the corresponding numerical results become unreliable, because the condition numbers are very large.
In the paper, MQM is proposed to calculate weakly singular integrals by Sidi quadrature rules , which makes the calculation of the discrete matrix becomes very simple and straightforward without any singular integrals. The convergence theory of approximations is given by estimating eigenvalues of the discrete matrix and using Anselone’s collectively compact convergent theory , which shows that the method retains the optimal convergence order and possesses the optimal condition number . Since MQM possesses the multivariate asymptotic expansion of errors, we can construct SEA to obtain the convergence order . Once discrete equations on some coarse meshes are solved in parallel, the accuracy of numerical solutions can be greatly improved by SEA.
This paper is organized as follows. Section 2 includes the singularity analysis of the integral kernels and the solution. In Section 3, the MQM is described. In Section 4, we can obtain multiparameter asymptotic expansion of errors and SEA is described. In Section 5, a numerical example is provided to verify the theoretical results.
2. The Integral Kernels and the Solution of Singularity Analysis
Define boundary integral operators on : Then (7) can be converted into a matrix operator equation where , , and .
Assume that can be described by the parameter mapping: with , . Using Sidi periodic transformation  where , then operator (8) is also converted into integral operator on : where and . Then (9) can be rewritten as where , , , , , and .
Let , , and be the kernels of the integral operators , , and , respectively. Then, the following results hold.(1) is a logarithmic singular function on and .(2) is a continuous function on and .(3)For , is a continuous function on and .(4)The solution is a smooth function under (11).
Lemma 1. If or and , then and are continuous on .
Proof. Without loss of generality, we assume that the origin of coordinates is a vertex with an interior angle . By (6), can be expressed by
where and . We assume that ; namely, has the logarithmic singularity at points and is continuously differentiable in . Consider
So the singularity of occurs in . In order to remove the singularity, we define
Since have 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, are continuous on . We can prove that are also continuous on . The proof of Lemma 1 is completed.
3. Mechanical Quadrature Method
Let be mesh widths and let be nodes. Since an integral operator is continuous, by the trapezoidal or the midpoint rule , we can construct Nyström approximate operator for the integral operator , defined by and the error Since has the singularities on , by Sidi quadrature formula , we get the following approximations of integral operator : and the error where is the derivative of the Riemann zeta function.
Then, we can get the approximate equation of (13): where , , , , , , , , , and . Obviously, (24) is a system of linear equations with unknowns. Once is solved by (24), the solution can be computed by is symmetric circular matrix and has the form of
Lemma 2 (see ). (1) There exists a positive so that the eigenvalues of satisfy . (2) The condition number of is . (3) is invertible, and is uniformly bounded with the spectral norm .
Based on Lemma 2, we immediately get the following corollary.
Corollary 3. (1) is invertible, and is uniformly bounded with the spectral norm . (2) The condition number of is , where .
Now we give the following definitions to discuss the approximate convergence in (27).
Define a subspace of the space with a norm . Let be a piecewise linear function subspace with base points , and let be base function satisfying . Define a prolongation operator satisfying Define a restricted operator satisfying
Replacing , , and , we construct an operator equation where and . Obviously, if is the solution of (30), then must be the solution of (27); conversely, if is the solution of (27), then must be the solution of (30). In order to prove the convergence of MQM, we give the following lemma.
Lemma 4 (see ). The operator sequence is uniformly bounded and convergent to the embedding operator .
Corollary 5. Let the Nyström approximation be defined by (20). (1) For , one has (2) For , one has where is the Nyström approximation of integral operator with the kernel and denotes the collectively compact convergence.
Theorem 6. Let with and be 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 , by Corollary 5, in , and there exists a convergent subsequence in . For , we have By Lemma 4, 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 converges. Similarly, it can be concluded that there exists an infinite subsequence such that converges. Therefore, is collectively compact convergent sequence, and , where shows the point convergence. We complete the proof.
For the stability of MQM, we have the following corollary.
Corollary 7. Let with , let be smooth curve, let , , and be the discrete matrices defined by (20) and (22), respectively, and let be the eigenvalues of discrete matrix . Then, there exists the bound of condition number where , , is the mesh step size of a curved edge .
4. Multiparameter Asymptotic Expansion of Errors and SEA
In this section, we derive the multivariate asymptotic expansion of solution errors and describe SEA. We first provide the main result.
Theorem 8. Let with . There exists a vector function independent of such that the following multiparameter asymptotic expansion holds at nodes: where , .
Proof. By (20) and (22), there exists the asymptotic expansion
where and with .
Using (13), (21), (23), and (24), we obtain where with and with . From Corollary 3, we can obtain Constructing the auxiliary equation and its approximate equation and substituting (42) into (40), we obtain Since is uniformly bounded from Theorem 6, we get Replacing with , we can complete the proof.
The multiparameter asymptotic expansion (37) means that SEA can be applied to solve (7); that is, higher order accuracy at coarse grid points can be obtained by solving some discrete equations in parallel. The process of SEA is as follows .
Step 1. Take and , and solve (24) under mesh parameters in parallel to get the numerical solutions and , , .
Step 2. Compute and , by (25), , and .
Step 3. Compute the extrapolation on the coarse grids as follows:
Step 4. Compute a posteriori error estimate on the coarse grids as follows: In the actual calculation process, a posteriori error estimate is immediately used to verify the calculation accuracy.
5. Numerical Example
In this section, we carry out a numerical example for the Helmholtz equation by MQM and SEA, in order to verify the error and stability analysis in the previous sections. Let be the real part of and let be the imaginary part of , where denotes a point. Post-error and SEA-error denote the a posteriori error and the error after SEA once, respectively.
Example 1. Consider Helmholtz equation with on a plate domain . We describe the boundary with , , , and . Dirichlet boundary conditions corresponding to the analytical solution were applied to the boundary. We compute the numerical solution with and using . The numerical results are listed in Tables 1 and 2. From Table 1, we can know that the convergence rates of are for MQM and that the convergence rates of are at least for SEA. From Table 2, we can see to indicate Corollary 7. It verifies the stability of convergent theory for MQM. Those results coincide with the theoretical analysis made.
6. Concluding Remarks
To close this paper, let us make a few concluding remarks.(1)Evaluation on entries of discrete matrices is very simple and straightforward without any singular integrals by MQM.(2)The numerical experiments show that MQM retains the optimal convergence order and possesses the optimal condition number which shows MQM owns the excellent stability. The approximate solutions accuracy order is at least after splitting extrapolation once, which is a great improvement in accuracy.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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