Numerical Solutions to Nonsmooth Dirichlet Problems Based on Lumped Mass Finite Element Discretization
We apply a lumped mass finite element to approximate Dirichlet problems for nonsmooth elliptic equations. It is proved that the lumped mass FEM approximation error in energy norm is the same as that of standard piecewise linear finite element approximation. Under the quasi-uniform mesh condition a...
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| Format: | Article |
| Language: | English |
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Wiley
2014-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2014/549305 |
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| _version_ | 1832556524498583552 |
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| author | Haixiong Yu Jinping Zeng |
| author_facet | Haixiong Yu Jinping Zeng |
| author_sort | Haixiong Yu |
| collection | DOAJ |
| description | We apply a lumped mass finite element to approximate Dirichlet problems for nonsmooth elliptic equations. It is proved that the lumped mass FEM approximation error in energy norm is the same as that of standard piecewise linear finite element approximation. Under the quasi-uniform mesh condition and the maximum angle condition, we show that the operator in the finite element problem is diagonally isotone and off-diagonally antitone. Therefore, some monotone convergent algorithms can be used. As an example, we prove that the nonsmooth Newton-like algorithm is convergent monotonically
if Gauss-Seidel iteration is used to solve the Newton's equations iteratively. Some numerical experiments are presented. |
| format | Article |
| id | doaj-art-c1a15dd38a7a46b1b2fd28da815bce90 |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2014-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-c1a15dd38a7a46b1b2fd28da815bce902025-02-03T05:45:12ZengWileyAbstract and Applied Analysis1085-33751687-04092014-01-01201410.1155/2014/549305549305Numerical Solutions to Nonsmooth Dirichlet Problems Based on Lumped Mass Finite Element DiscretizationHaixiong Yu0Jinping Zeng1College of Science, Nanchang Institute of Technology, Nanchang 330099, ChinaCollege of Computer, Dongguan University of Technology, Dongguan 523000, ChinaWe apply a lumped mass finite element to approximate Dirichlet problems for nonsmooth elliptic equations. It is proved that the lumped mass FEM approximation error in energy norm is the same as that of standard piecewise linear finite element approximation. Under the quasi-uniform mesh condition and the maximum angle condition, we show that the operator in the finite element problem is diagonally isotone and off-diagonally antitone. Therefore, some monotone convergent algorithms can be used. As an example, we prove that the nonsmooth Newton-like algorithm is convergent monotonically if Gauss-Seidel iteration is used to solve the Newton's equations iteratively. Some numerical experiments are presented.http://dx.doi.org/10.1155/2014/549305 |
| spellingShingle | Haixiong Yu Jinping Zeng Numerical Solutions to Nonsmooth Dirichlet Problems Based on Lumped Mass Finite Element Discretization Abstract and Applied Analysis |
| title | Numerical Solutions to Nonsmooth Dirichlet Problems Based on Lumped Mass Finite Element Discretization |
| title_full | Numerical Solutions to Nonsmooth Dirichlet Problems Based on Lumped Mass Finite Element Discretization |
| title_fullStr | Numerical Solutions to Nonsmooth Dirichlet Problems Based on Lumped Mass Finite Element Discretization |
| title_full_unstemmed | Numerical Solutions to Nonsmooth Dirichlet Problems Based on Lumped Mass Finite Element Discretization |
| title_short | Numerical Solutions to Nonsmooth Dirichlet Problems Based on Lumped Mass Finite Element Discretization |
| title_sort | numerical solutions to nonsmooth dirichlet problems based on lumped mass finite element discretization |
| url | http://dx.doi.org/10.1155/2014/549305 |
| work_keys_str_mv | AT haixiongyu numericalsolutionstononsmoothdirichletproblemsbasedonlumpedmassfiniteelementdiscretization AT jinpingzeng numericalsolutionstononsmoothdirichletproblemsbasedonlumpedmassfiniteelementdiscretization |