Series solution of Laplace problems
This is a tutorial introduction to a known method, offering no new results or improvements for practitioners.
The paper presents a tutorial on solving Laplace problems in multiply connected domains using series expansions with least-squares fitting on the boundary, achieving exponential convergence for well-behaved boundary data, demonstrated on a Cantor set with up to 2048 components.
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.