On the solution of elliptic free-boundary problems via Newton's method

The performance of a numerical technique for locating the free surface without mesh iteration is investigated in the context of steady state pore fluid diffusion through porous and deformable media. The free boundary is located from a fixed mesh solution in which the required constraints are imposed by penalization of the variational equation. Central to the efficiency of the algorithm is the use of a ‘relaxed penalty function’ which makes it possible to linearize the penalized equation consistently. The solution is shown to converge orders of magnitude faster than some of the most widely used algorithms for solving free-boundary problems associated with flow of fluids through porous media.