The numerical solution of elliptic free boundary problems using multigrid techniques

Abstract

A multigrid algorithm is developed for the numerical solution of elliptic free boundary problems. The domain of the problem is mapped onto a rectangle and the governing equations discretized using finite differences. The resulting algebraic system is solved iteratively using a multigrid V cycle. For a convergent relaxation procedure it is necessary to use line iteration perpendicular to the free boundary simultaneously altering the values of the dependent variable and the position of the boundary, which is conveniently done using a single Newton iteration. Three problems are considered, a Poisson type problem, a steady state heat transfer problem, and one from electrochemical machining. The first two problems rapidly converge in a few multigrid cycles, the third converges less rapidly though adequately. Since full multigrid (FMG) is used, the results on the three finest grids could be combined to give accurate results of sixth order.