Series solution of Laplace problems


  • Lloyd Nicholas Trefethen, FRS University of Oxford Mathematical Institute



Green function, conformal mapping, harmonic measure, Laplace problem, series expansion, least-squares, Cantor set.


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. doi:10.1017/S1446181118000093

Author Biography

Lloyd Nicholas Trefethen, FRS, University of Oxford Mathematical Institute

Fellow of the Royal Society Professor of Numerical Analysis, Oxford University Fellow of Balliol College Head of Oxford's Numerical Analysis Group





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