Duality-based domain decomposition with natural coarse-space for variational inequalities

An efficient non-overlapping domain decomposition algorithm of Neumann–Neumann type for solving variational inequalities arising from the elliptic boundary value problems with inequality boundary conditions has been presented. The discretized problem is first turned by the duality theory of convex programming into a quadratic programming problem with bound and equality constraints and the latter is further modified by means of orthogonal projectors to the natural coarse space introduced recently by Farhat and Roux. The resulting problem is then solved by an augmented Lagrangian type algorithm with an outer loop for the Lagrange multipliers for the equality constraints and an inner loop for the solution of the bound constrained quadratic programming problems. The projectors are shown to guarantee an optimal rate of convergence of iterative solution of auxiliary linear problems. Reported theoretical results and numerical experiments indicate high numerical and parallel scalability of the algorithm.