A multigrid local smoother approach for a domain decomposition solver over non-matching grids

In this paper we consider a multigrid approach for solving elliptic equations over non-matching grids with domain decomposition methods. The domain is partitioned into subdomains with different mesh levels that do not match at the interface. The proposed algorithm searches for the global solution over different levels by projecting the residuals on the overlap region. This method is used in conjunction with a domain decomposition solver which only requires, in each iteration step, the solutions of several small local subproblems over finite element blocks. This algorithm is shown to converge to the solution of the corresponding Lagrange multiplier problem for non-matching grids. The convergence properties of the algorithms are analyzed and numerical examples are presented. When the multigrid and domain decomposition approaches are combined, the method is shown to be reliable and easy to implement. Furthermore the local nature of the solver allows for a straightforward implementation on multiple parallel computers and graphics processing unit (GPU) clusters.