Primal–dual weak Galerkin finite element methods for elliptic Cauchy problems
The authors propose and analyze a well-posed numerical scheme for a type of ill-posed elliptic Cauchy problem by using a constrained minimization approach combined with the weak Galerkin finite element method. The resulting Euler–Lagrange formulation yields a system of equations involving the origin...
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| Published in | Computers & mathematics with applications (1987) Vol. 79; no. 3; pp. 746 - 763 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Oxford
Elsevier Ltd
01.02.2020
Elsevier BV |
| Subjects | |
| Online Access | Get full text |
| ISSN | 0898-1221 1873-7668 |
| DOI | 10.1016/j.camwa.2019.07.031 |
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| Abstract | The authors propose and analyze a well-posed numerical scheme for a type of ill-posed elliptic Cauchy problem by using a constrained minimization approach combined with the weak Galerkin finite element method. The resulting Euler–Lagrange formulation yields a system of equations involving the original equation for the primal variable and its adjoint for the dual variable, and is thus an example of the primal–dual weak Galerkin finite element method. This new primal–dual weak Galerkin algorithm is consistent in the sense that the system is symmetric, well-posed, and is satisfied by the exact solution. A certain stability and error estimates were derived in discrete Sobolev norms, including one in a weak L2 topology. Some numerical results are reported to illustrate and validate the theory developed in the paper. |
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| AbstractList | The authors propose and analyze a well-posed numerical scheme for a type of ill-posed elliptic Cauchy problem by using a constrained minimization approach combined with the weak Galerkin finite element method. The resulting Euler–Lagrange formulation yields a system of equations involving the original equation for the primal variable and its adjoint for the dual variable, and is thus an example of the primal–dual weak Galerkin finite element method. This new primal–dual weak Galerkin algorithm is consistent in the sense that the system is symmetric, well-posed, and is satisfied by the exact solution. A certain stability and error estimates were derived in discrete Sobolev norms, including one in a weak L2 topology. Some numerical results are reported to illustrate and validate the theory developed in the paper. |
| Author | Wang, Junping Wang, Chunmei |
| Author_xml | – sequence: 1 givenname: Chunmei orcidid: 0000-0001-5181-8265 surname: Wang fullname: Wang, Chunmei email: chunmei.wang@ttu.edu organization: Department of Mathematics, Texas Tech University, Lubbock, TX 79409, USA – sequence: 2 givenname: Junping surname: Wang fullname: Wang, Junping email: jwang@nsf.gov organization: Division of Mathematical Sciences, National Science Foundation, Alexandria, VA 22314, USA |
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| Title | Primal–dual weak Galerkin finite element methods for elliptic Cauchy problems |
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