Series solution of Laplace problems
At the ANZIAM conference in Hobart in February, 2018, there were several talks on the solution of Laplace problems in multiply connected domains by means of conformal mapping. It appears to be not widely known that such problems can also be solved by the elementary method of series expansions with c...
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| Format: | Journal article |
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Cambridge University Press
2018
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| _version_ | 1826272285754916864 |
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| author | Trefethen, LN |
| author_facet | Trefethen, LN |
| author_sort | Trefethen, LN |
| collection | OXFORD |
| description | At the ANZIAM conference in Hobart in February, 2018, there were several talks on the solution of Laplace problems in multiply connected domains by means of conformal mapping. It appears to be not widely known that such problems can also be solved by the elementary method of series expansions with coefficients determined by least-squares fitting on the boundary. (These are not convergent series; the coefficients depend on the degree of the approximation.) Here we give a tutorial introduction to this method, which converges at an exponential rate if the boundary data are sufficiently well-behaved. The mathematical foundations go back to Runge in 1885 and Walsh in 1929. One of our examples involves an approximate Cantor set with up to 2048 components. |
| first_indexed | 2024-03-06T22:10:08Z |
| format | Journal article |
| id | oxford-uuid:5187c78e-2929-4b3a-bc3b-2fa8aeb7dfbf |
| institution | University of Oxford |
| last_indexed | 2024-03-06T22:10:08Z |
| publishDate | 2018 |
| publisher | Cambridge University Press |
| record_format | dspace |
| spelling | oxford-uuid:5187c78e-2929-4b3a-bc3b-2fa8aeb7dfbf2022-03-26T16:20:04ZSeries solution of Laplace problemsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:5187c78e-2929-4b3a-bc3b-2fa8aeb7dfbfSymplectic Elements at OxfordCambridge University Press2018Trefethen, LNAt the ANZIAM conference in Hobart in February, 2018, there were several talks on the solution of Laplace problems in multiply connected domains by means of conformal mapping. It appears to be not widely known that such problems can also be solved by the elementary method of series expansions with coefficients determined by least-squares fitting on the boundary. (These are not convergent series; the coefficients depend on the degree of the approximation.) Here we give a tutorial introduction to this method, which converges at an exponential rate if the boundary data are sufficiently well-behaved. The mathematical foundations go back to Runge in 1885 and Walsh in 1929. One of our examples involves an approximate Cantor set with up to 2048 components. |
| spellingShingle | Trefethen, LN Series solution of Laplace problems |
| title | Series solution of Laplace problems |
| title_full | Series solution of Laplace problems |
| title_fullStr | Series solution of Laplace problems |
| title_full_unstemmed | Series solution of Laplace problems |
| title_short | Series solution of Laplace problems |
| title_sort | series solution of laplace problems |
| work_keys_str_mv | AT trefethenln seriessolutionoflaplaceproblems |