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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| Format: | Article |
| Language: | English |
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Wiley
2014-01-01
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| Series: | Journal of Applied Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2014/812505 |
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| author | Hu Li Yanying Ma |
| author_facet | Hu Li Yanying Ma |
| author_sort | Hu Li |
| collection | DOAJ |
| 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(hmax5) by splitting extrapolation algorithm (SEA). A numerical example is provided to support our theoretical analysis. |
| format | Article |
| id | doaj-art-ebe883505a0e42ba94a860e8b30d6331 |
| institution | OA Journals |
| issn | 1110-757X 1687-0042 |
| language | English |
| publishDate | 2014-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Applied Mathematics |
| spelling | doaj-art-ebe883505a0e42ba94a860e8b30d63312025-08-20T02:21:49ZengWileyJournal 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(hmax5) by splitting extrapolation algorithm (SEA). A numerical example is provided to support our theoretical analysis.http://dx.doi.org/10.1155/2014/812505 |
| 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 |
| title | 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_short | 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 |
| url | http://dx.doi.org/10.1155/2014/812505 |
| work_keys_str_mv | AT huli mechanicalquadraturemethodandsplittingextrapolationforsolvingdirichletboundaryintegralequationofhelmholtzequationonpolygons AT yanyingma mechanicalquadraturemethodandsplittingextrapolationforsolvingdirichletboundaryintegralequationofhelmholtzequationonpolygons |