Mechanical Quadrature Method and Splitting Extrapolation for Solving Dirichlet Boundary Integral Equation of Helmholtz Equation on Polygons. (10th July 2014)

• Record Type: Journal Article
• Authors: Li, Hu; Ma, Yanying
• Other Names: Yu Bo Academic Editor.
• Is Part Of: Journal of applied mathematics. Volume 2014(2014)
• Publication Date: 2014-07-10
• Subjects: Mathematics -- Periodicals; 519.05
• DOI: 10.1155/2014/812505
• Languages: English
• ISSNs: 1110-757X

Abstract

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 Γ m ( m = 1, …, 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.