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 themechanical quadraturemethod (MQM) using specific quadrature rule to deal with weakly singular integrals. Denote by h m themeshwidth of a curved edge Γ m (m = 1, . . . , d) of polygons. Then, the multivariate asymptotic error expansion of MQM accompanied with O(h3 m ) for all mesh widths h m is obtained. Hence, once discrete equations with coarse meshes are solved in parallel, the higher accuracy order of numerical approximations can be at least O(h5 max) by splitting extrapolation algorithm (SEA). A numerical example is provided to support our theoretical analysis.

And  (1)  0 =  0 + 0 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  = 0.57721 . . . is Euler constant.
In what follows, in order to analyze properties of the kernel, we decompose the kernel Equation ( 7) is weakly singular BIE system of the first kind, whose solution exists and is unique as   ̸ = 1 [1], where   is the logarithmic capacity (i.e., the transfinite diameter).Once V() is solved from (7), the function ()( ∈ Ω) 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 [4] 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 [5], 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 [6], which shows that the method retains the optimal convergence order (ℎ 3 max ) and possesses the optimal condition number (ℎ −1 min ).Since MQM possesses the multivariate asymptotic expansion of errors, we can construct SEA to obtain the convergence order (ℎ 5 max ).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.
Then, we can get the approximate equation of ( 13): where ℎ 0 is symmetric circular matrix and has the form of Lemma 2 (see [9]).From Corollary 3, we know that (24) is equivalent to where  ℎ denotes the unit matrix.Now we give the following definitions to discuss the approximate convergence in (27).
Obviously, if Vℎ is the solution of (30), then  ℎ  ℎ must be the solution of (27); conversely, if  ℎ is the solution of (27), then  ℎ  ℎ must be the solution of (30).In order to prove the convergence of MQM, we give the following lemma.
We firstly consider the first component of  ℎ ( For Γ  ∩ Γ 1 = 0, by Corollary 5,  1 ( ) ℎ }.Based on the above two cases it is proved that there exists an infinite subsequence  (1) ⊂  such that the first component converges.Similarly, it can be concluded that there exists an infinite subsequence  () ⊂  (−1) ⊂ ⋅ ⋅ ⋅ ⊂  (1) ⊂  such that  ℎ ( where   → shows the point convergence.We complete the proof.
For the stability of MQM, we have the following corollary.

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.
The multiparameter asymptotic expansion (37) means that SEA can be applied to solve (7); that is, higher order accuracy (ℎ 3 0 ) at coarse grid points can be obtained by solving some discrete equations in parallel.The process of SEA is as follows [10].
Step 3. Compute the extrapolation on the coarse grids as follows: Step In the actual calculation process, a posteriori error estimate is immediately used to verify the calculation accuracy.

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.were applied to the boundary.We compute the numerical solution  ℎ with  = (0.5,0.5) and  = (0.6,0.6) using  3 ().The numerical results are listed in Tables 1 and 2.
From Table 1, we can know that the convergence rates of  ℎ are (ℎ 3 max ) for MQM and that the convergence rates of  ℎ are at least (ℎ 5 max ) for SEA.From

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 (ℎ 3 max ) and possesses the optimal condition number (ℎ −1 min ) which shows MQM owns the excellent stability.The approximate solutions accuracy order is at least (ℎ 5 max ) after splitting extrapolation once, which is a great improvement in accuracy.