An efficient iterative scheme for series solutions to Laplacian free boundary problems

Abstract

Non-linear Laplacian free boundary problems arise in many places in the physical sciences and engineering. Typical applications include locating the water table in groundwater problems, and fully non-linear problems such as flow over topography. Analytic series methods are used to solve these problems by iteratively improving an initial estimate of the free boundary location --- at each step, the problem reduces to solving a known boundary problem. As the boundary geometry is not regular, the series coefficients at each iteration are obtained by solving a matrix equation, instead of using an orthogonality relationship. The components of the matrix equations are inner products that result from minimising the boundary errors in the least squares (or $L_2$ norm) sense. As the size of the (normal) matrices generated are relatively small, most of the computational effort is spent evaluating these inner products. In this paper, an efficient method is presented to evaluate these integrals that result in an order of magnitude increase in the overall efficiency of the solution process. This increase in efficiency does not come at the cost of accuracy, after suitable modifications are made to the iterative process.