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SERIES SOLUTION OF LAPLACE PROBLEMS

Part of: Two-dimensional theory Elliptic equations and systems

Published online by Cambridge University Press:  06 July 2018

LLOYD N. TREFETHEN
LLOYD N. TREFETHEN*
Affiliation:
Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK email [email protected]
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Abstract

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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.

Keywords

Green functionconformal mappingharmonic measureLaplace problemseries expansionleast-squaresCantor set

MSC classification

Primary: 31A99: None of the above, but in this section
Secondary: 35J08: Green's functions
Type
Research Article
Information
The ANZIAM Journal , Volume 60 , Issue 1 , July 2018 , pp. 1 - 26
Copyright
© 2018 Australian Mathematical Society 

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