title: Accurate and efficient numerical solutions for elliptic obstacle problems.
abstract: 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.
authors.affiliation: Department of Mathematics, Sogang University, Ricci Building R1416, 35 Baekbeom-ro, Mapo-gu, Seoul, 04107 South Korea.
authors.forename: Philku
authors.initials: P
authors.lastname: Lee
authors.affiliation: Centennial Christian School International, 20 Shin Heung Ro 26-Gil, Yongsan Gu, Seoul, 140-833 South Korea.
authors.forename: Tai Wan
authors.initials: TW
authors.lastname: Kim
authors.affiliation: Mississippi State University, Mississippi State, MS 39762-5921 USA.
authors.forename: Seongjai
authors.initials: S
authors.lastname: Kim
keywords: elliptic obstacle problem; gradient-weighting method; obstacle relaxation; subgrid finite difference (FD); successive over-relaxation (SOR) method