Mechanical quadrature method and splitting extrapolation for solving Dirichlet boundary integral equation of Helmholtz equation on polygons. (English) Zbl 1442.65459

Li, Hu; Ma, Yanying

J. Appl. Math. 2014, Article ID 812505, 7 p. (2014).

Summary: 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 \(\Gamma_m\) (\(m=1,\dots,d\)) of polygons. Then, the multivariate asymptotic error expansion of MQM accompanied with \(O(h_m^3)\) 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 \(O(h_{\max}^5)\) by splitting extrapolation algorithm (SEA). A numerical example is provided to support our theoretical analysis.

MSC:

65R20	Numerical methods for integral equations
45E10	Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
65D32	Numerical quadrature and cubature formulas

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