Versions of the Collocation and Least Residuals Method for Solving Problems of Mathematical Physics in the Convex Quadrangular Domains
The new versions of the collocations and least residuals (CLR) method of high-order accuracy are proposed and implemented for the numerical solution of the boundary value problems for PDE in the convex quadrangular domains. Their implementation and numerical experiments are performed by the examples...
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| Language: | English |
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Yaroslavl State University
2017-10-01
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| Series: | Моделирование и анализ информационных систем |
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| Online Access: | https://www.mais-journal.ru/jour/article/view/585 |
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| author | Vasily A. Belyaev Vasily P. Shapeev |
| author_facet | Vasily A. Belyaev Vasily P. Shapeev |
| author_sort | Vasily A. Belyaev |
| collection | DOAJ |
| description | The new versions of the collocations and least residuals (CLR) method of high-order accuracy are proposed and implemented for the numerical solution of the boundary value problems for PDE in the convex quadrangular domains. Their implementation and numerical experiments are performed by the examples of solving the biharmonic and Poisson equations. The solution of the biharmonic equation is used for simulation of the stress-strain state of an isotropic plate under the action of the transverse load. Differential problems are projected into the space of fourth-degree polynomials by the CLR method. The boundary conditions for the approximate solution are put down exactly on the boundary of the computational domain. The versions of the CLR method are implemented on the grids, which are constructed by two different ways. In the first version, a “quasiregular” grid is constructed in the domain, the extreme lines of this grid coincide with the boundaries of the domain. In the second version, the domain is initially covered by a regular grid with rectangular cells. Herewith, the collocation and matching points that are situated outside the domain are used for approximation of the differential equations in the boundary cells that had been crossed by the boundary. In addition the “small” irregular triangular cells that had been cut off by the domain boundary from rectangular cells of the initial regular grid are joined to adjacent quadrangular cells. This technique allowed to essentially reduce the conditionality of the system of linear algebraic equations of the approximate problem in comparison with the case when small irregular cells together with other cells were used as independent ones for constructing an approximate solution of the problem. It is shown that the approximate solution of problems converges with high order and matches with high accuracy with the analytical solution of the test problems in the case of the known solution in numerical experiments on the convergence of the solution of various problems on a sequence of grids. |
| format | Article |
| id | doaj-art-25a9abd9147b4e9b802bafc088f4b291 |
| institution | Kabale University |
| issn | 1818-1015 2313-5417 |
| language | English |
| publishDate | 2017-10-01 |
| publisher | Yaroslavl State University |
| record_format | Article |
| series | Моделирование и анализ информационных систем |
| spelling | doaj-art-25a9abd9147b4e9b802bafc088f4b2912025-08-20T03:37:50ZengYaroslavl State UniversityМоделирование и анализ информационных систем1818-10152313-54172017-10-0124562964810.18255/1818-1015-2017-5-629-648427Versions of the Collocation and Least Residuals Method for Solving Problems of Mathematical Physics in the Convex Quadrangular DomainsVasily A. Belyaev0Vasily P. Shapeev1Khristianovich Institute of Theoretical and Applied Mechanics, Siberian Branch of the Russian Academy of SciencesKhristianovich Institute of Theoretical and Applied Mechanics, Siberian Branch of the Russian Academy of Sciences; Novosibirsk National Research UniversityThe new versions of the collocations and least residuals (CLR) method of high-order accuracy are proposed and implemented for the numerical solution of the boundary value problems for PDE in the convex quadrangular domains. Their implementation and numerical experiments are performed by the examples of solving the biharmonic and Poisson equations. The solution of the biharmonic equation is used for simulation of the stress-strain state of an isotropic plate under the action of the transverse load. Differential problems are projected into the space of fourth-degree polynomials by the CLR method. The boundary conditions for the approximate solution are put down exactly on the boundary of the computational domain. The versions of the CLR method are implemented on the grids, which are constructed by two different ways. In the first version, a “quasiregular” grid is constructed in the domain, the extreme lines of this grid coincide with the boundaries of the domain. In the second version, the domain is initially covered by a regular grid with rectangular cells. Herewith, the collocation and matching points that are situated outside the domain are used for approximation of the differential equations in the boundary cells that had been crossed by the boundary. In addition the “small” irregular triangular cells that had been cut off by the domain boundary from rectangular cells of the initial regular grid are joined to adjacent quadrangular cells. This technique allowed to essentially reduce the conditionality of the system of linear algebraic equations of the approximate problem in comparison with the case when small irregular cells together with other cells were used as independent ones for constructing an approximate solution of the problem. It is shown that the approximate solution of problems converges with high order and matches with high accuracy with the analytical solution of the test problems in the case of the known solution in numerical experiments on the convergence of the solution of various problems on a sequence of grids.https://www.mais-journal.ru/jour/article/view/585collocations and least residuals methodboundary value problemnon-canonical domainirregular gridhigh order approximationpoisson’s equationbiharmonic equation |
| spellingShingle | Vasily A. Belyaev Vasily P. Shapeev Versions of the Collocation and Least Residuals Method for Solving Problems of Mathematical Physics in the Convex Quadrangular Domains Моделирование и анализ информационных систем collocations and least residuals method boundary value problem non-canonical domain irregular grid high order approximation poisson’s equation biharmonic equation |
| title | Versions of the Collocation and Least Residuals Method for Solving Problems of Mathematical Physics in the Convex Quadrangular Domains |
| title_full | Versions of the Collocation and Least Residuals Method for Solving Problems of Mathematical Physics in the Convex Quadrangular Domains |
| title_fullStr | Versions of the Collocation and Least Residuals Method for Solving Problems of Mathematical Physics in the Convex Quadrangular Domains |
| title_full_unstemmed | Versions of the Collocation and Least Residuals Method for Solving Problems of Mathematical Physics in the Convex Quadrangular Domains |
| title_short | Versions of the Collocation and Least Residuals Method for Solving Problems of Mathematical Physics in the Convex Quadrangular Domains |
| title_sort | versions of the collocation and least residuals method for solving problems of mathematical physics in the convex quadrangular domains |
| topic | collocations and least residuals method boundary value problem non-canonical domain irregular grid high order approximation poisson’s equation biharmonic equation |
| url | https://www.mais-journal.ru/jour/article/view/585 |
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