Mechanical Quadrature Method and Splitting Extrapolation for Solving Dirichlet Boundary Integral Equation of Helmholtz Equation on Polygons

Abstract

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 $h_m$ the mesh width of a curved edge ${\Gamma }_m\hspace{0.17em}\hspace{0.17em}(m=\mathrm{1},\dots ,d)$ of polygons. Then, the multivariate asymptotic error expansion of MQM accompanied with $O(h_m^{\mathrm{3}})$ 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 $O(h_{\max }^{\mathrm{5}})$ by splitting extrapolation algorithm (SEA). A numerical example is provided to support our theoretical analysis.