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.
Highlights
Variational inequalities have been extensively studied as one of key issues in calculus of variations and in the applied sciences
For the examples presented in this subsection, the numerical solutions are solved as follows: (a) the problem is solved with ε =, (b) the free boundary is estimated with (k, k ) = (, ) and subgrid finite difference (FD) schemes are applied at neighboring grid points as in Section, and (c) another round of iterations is applied to satisfy the tolerance ε =
Various numerical algorithms have been suggested for solving elliptic obstacle problems effectively, most of the algorithms presented in the literature are yet to be improved for both accuracy and efficiency
Summary
Variational inequalities have been extensively studied as one of key issues in calculus of variations and in the applied sciences. The above example has motivated the authors to develop an effective numerical algorithm for elliptic obstacle problems in D which detects the neighboring set of the free boundary, determines the subgrid proportions (r’s), and updates the solution for an improved accuracy using subgrid FD schemes. Variants of Algorithm NJ for the GS and the SOR can be formulated as for the linear obstacle problem; they would be denoted respectively by NGS and NSOR(ω). Such symmetric coercive optimization problems, the SOR methods are much more efficient than the Jacobi and Gauss-Seidel methods. We will exploit LSOR(ω) and NSOR(ω) for numerical comparisons with state-of-the-art methods, by setting the relaxation parameter ω optimal
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