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

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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