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

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 the mechanical quadrature method (MQM) using specific quadrature rule to deal with weakly singular integr...

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Main Authors: Hu Li, Yanying Ma
Format: Article
Language:English
Published: Hindawi Limited 2014-01-01
Series:Journal of Applied Mathematics
Online Access:http://dx.doi.org/10.1155/2014/812505
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spelling doaj-ebe883505a0e42ba94a860e8b30d63312020-11-24T23:15:35ZengHindawi LimitedJournal of Applied Mathematics1110-757X1687-00422014-01-01201410.1155/2014/812505812505Mechanical Quadrature Method and Splitting Extrapolation for Solving Dirichlet Boundary Integral Equation of Helmholtz Equation on PolygonsHu Li0Yanying Ma1School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, ChinaSchool of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, ChinaWe 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 hm the mesh width of a curved edge Γm  (m=1,…,d) of polygons. Then, the multivariate asymptotic error expansion of MQM accompanied with O(hm3) for all mesh widths hm 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(hmax⁡5) by splitting extrapolation algorithm (SEA). A numerical example is provided to support our theoretical analysis.http://dx.doi.org/10.1155/2014/812505
collection DOAJ
language English
format Article
sources DOAJ
author Hu Li
Yanying Ma
spellingShingle Hu Li
Yanying Ma
Mechanical Quadrature Method and Splitting Extrapolation for Solving Dirichlet Boundary Integral Equation of Helmholtz Equation on Polygons
Journal of Applied Mathematics
author_facet Hu Li
Yanying Ma
author_sort Hu Li
title Mechanical Quadrature Method and Splitting Extrapolation for Solving Dirichlet Boundary Integral Equation of Helmholtz Equation on Polygons
title_short Mechanical Quadrature Method and Splitting Extrapolation for Solving Dirichlet Boundary Integral Equation of Helmholtz Equation on Polygons
title_full Mechanical Quadrature Method and Splitting Extrapolation for Solving Dirichlet Boundary Integral Equation of Helmholtz Equation on Polygons
title_fullStr Mechanical Quadrature Method and Splitting Extrapolation for Solving Dirichlet Boundary Integral Equation of Helmholtz Equation on Polygons
title_full_unstemmed Mechanical Quadrature Method and Splitting Extrapolation for Solving Dirichlet Boundary Integral Equation of Helmholtz Equation on Polygons
title_sort mechanical quadrature method and splitting extrapolation for solving dirichlet boundary integral equation of helmholtz equation on polygons
publisher Hindawi Limited
series Journal of Applied Mathematics
issn 1110-757X
1687-0042
publishDate 2014-01-01
description 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 hm the mesh width of a curved edge Γm  (m=1,…,d) of polygons. Then, the multivariate asymptotic error expansion of MQM accompanied with O(hm3) for all mesh widths hm 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(hmax⁡5) by splitting extrapolation algorithm (SEA). A numerical example is provided to support our theoretical analysis.
url http://dx.doi.org/10.1155/2014/812505
work_keys_str_mv AT huli mechanicalquadraturemethodandsplittingextrapolationforsolvingdirichletboundaryintegralequationofhelmholtzequationonpolygons
AT yanyingma mechanicalquadraturemethodandsplittingextrapolationforsolvingdirichletboundaryintegralequationofhelmholtzequationonpolygons
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