Two-level Schwarz method for unilateral variational inequalities

The numerical solution of variational inequalities of obstacle type associated with second-order elliptic operators is considered. Iterative methods based on the domain decomposition approach are proposed for discrete obstacle problems arising from the continuous, piecewise linear finite element approximation of the differential problem. A new variant of the Schwarz methodology, called the two-level Schwarz method, is developed offering the possibility of making use of fast linear solvers (e.g., linear multigrid and fictitious domain methods) for the genuinely nonlinear obstacle problems. Namely, by using particular monotonicity results, the computational domain can be partitioned into (mesh) subdomains with linear and nonlinear (obstacle-type) subproblems. By taking advantage of this domain decomposition and fast linear solvers, efficient implementation algorithms for large-scale discrete obstacle problems can be developed. The last part of the paper is devoted to illustrative numerical experiments.