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
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.