Accurate and efficient numerical solutions for elliptic obstacle problems

Elliptic obstacle problems are formulated to find either superharmonic solutions or minimal surfaces that lie on or over the obstacles, by incorporating inequality constraints. In order to solve such problems effectively using finite difference (FD) methods, the article investigates simple iterative algorithms based on the successive over-relaxation (SOR) method. It introduces subgrid FD methods to reduce the accuracy deterioration occurring near the free boundary when the mesh grid does not match with the free boundary. For nonlinear obstacle problems, a method of gradient-weighting is introduced to solve the problem more conveniently and efficiently. The iterative algorithm is analyzed for convergence for both linear and nonlinear obstacle problems. An effective strategy is also suggested to find the optimal relaxation parameter. It has been numerically verified that the resulting obstacle SOR iteration with the optimal parameter converges about one order faster than state-of-the-art methods and the subgrid FD methods reduce numerical errors by one order of magnitude, for most cases. Various numerical examples are given to verify the claim.