Solving Laplace problems with corner singularities via rational functions
A new method is introduced for solving Laplace problems on 2D regions with corners by approximation of boundary data by the real part of a rational function with fixed poles exponentially clustered near each corner. Greatly extending a result of D. J. Newman in 1964 in approximation theory, we first...
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| Format: | Journal article |
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Society for Industrial and Applied Mathematics
2019
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| _version_ | 1797062580865335296 |
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| author | Gopal, A Trefethen, L |
| author_facet | Gopal, A Trefethen, L |
| author_sort | Gopal, A |
| collection | OXFORD |
| description | A new method is introduced for solving Laplace problems on 2D regions with corners by approximation of boundary data by the real part of a rational function with fixed poles exponentially clustered near each corner. Greatly extending a result of D. J. Newman in 1964 in approximation theory, we first prove that such approximations can achieve root-exponential convergence for a wide range of problems, all the way up to the corner singularities. We then develop a numerical method to compute approximations via linear least-squares fitting on the boundary. Typical problems are solved in < 1s on a laptop to 8-digit accuracy, with the accuracy guaranteed in the interior by the maximum principle. The computed solution is represented globally by a single formula, which can be evaluated in tens of microseconds at each point. |
| first_indexed | 2024-03-06T20:47:33Z |
| format | Journal article |
| id | oxford-uuid:366e6e2b-a6c9-42e1-b2bd-1d77c1dc2a6e |
| institution | University of Oxford |
| last_indexed | 2024-03-06T20:47:33Z |
| publishDate | 2019 |
| publisher | Society for Industrial and Applied Mathematics |
| record_format | dspace |
| spelling | oxford-uuid:366e6e2b-a6c9-42e1-b2bd-1d77c1dc2a6e2022-03-26T13:37:50ZSolving Laplace problems with corner singularities via rational functionsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:366e6e2b-a6c9-42e1-b2bd-1d77c1dc2a6eSymplectic Elements at OxfordSociety for Industrial and Applied Mathematics2019Gopal, ATrefethen, LA new method is introduced for solving Laplace problems on 2D regions with corners by approximation of boundary data by the real part of a rational function with fixed poles exponentially clustered near each corner. Greatly extending a result of D. J. Newman in 1964 in approximation theory, we first prove that such approximations can achieve root-exponential convergence for a wide range of problems, all the way up to the corner singularities. We then develop a numerical method to compute approximations via linear least-squares fitting on the boundary. Typical problems are solved in < 1s on a laptop to 8-digit accuracy, with the accuracy guaranteed in the interior by the maximum principle. The computed solution is represented globally by a single formula, which can be evaluated in tens of microseconds at each point. |
| spellingShingle | Gopal, A Trefethen, L Solving Laplace problems with corner singularities via rational functions |
| title | Solving Laplace problems with corner singularities via rational functions |
| title_full | Solving Laplace problems with corner singularities via rational functions |
| title_fullStr | Solving Laplace problems with corner singularities via rational functions |
| title_full_unstemmed | Solving Laplace problems with corner singularities via rational functions |
| title_short | Solving Laplace problems with corner singularities via rational functions |
| title_sort | solving laplace problems with corner singularities via rational functions |
| work_keys_str_mv | AT gopala solvinglaplaceproblemswithcornersingularitiesviarationalfunctions AT trefethenl solvinglaplaceproblemswithcornersingularitiesviarationalfunctions |