Two - Level Schwarz Method for Unilateralvariational Inequalitiesp

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 here proposed for discrete obstacle problems arising from the continuous, piecewise linear nite element approximation of the diierential problem. A new variant of the Schwarz methodology, called the two-level Schwarz method is developed ooering the possibility to make use of fast linear solvers (e.g., linear multigrid and ctitious 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, eecient implementation algorithms for large-scale discrete obstacle problems can be developed. The last part of the paper is devoted to the illustrative numerical experiments.