A comparison of analytic series methods for Laplacian free boundary problems

Solutions to Laplace's equation are required for a wide range of problems. Arguably, the most difficult class of problems involves a “free” boundary, where the location of one (or more) of the boundaries is initially unknown. Analytical solutions for these problems were restricted to regular boundary geometries. However, recently the classical series method has been modified, to cater for arbitrary boundary geometries, using least squares methods. For free boundary problems, solutions can be obtained by solving a sequence of known boundary problems—at each iteration, the series coefficients can be estimated. Efficient calculation of the series coefficients becomes very important, particularly when the number of iterations is relatively high. In this paper, three methods for estimating the series coefficients will be described, in the context of a free boundary problem. The computational cost of each method will be analysed and compared, and the most appropriate method for this class of problem is indicated.