Accelerated domain decomposition iterative procedures for mixed methods based on Robin transmission conditions

Abstract

A nonoverlapping domain decomposition iterative procedure based on Robin transmission conditions applicable to elliptic boundary problems was first introduced by P.~L.~Lions and later discussed by a number of authors under the assumption that the weighting of the flux and the trace of the solution in the Robin interface condition be independent of the iterative step number. Recently, the authors [6] studied a finite difference method for a Dirichlet problem and introduced a cycle of weights for the flux in this interface condition and proved that an acceleration in the convergence rate similar to that occurring for alternating-direction iteration using a cycle of pseudo-time steps results. The objects of this paper are to describe an analogous procedure for a mixed finite element approximation for a model Neumann problem and to consider an overlapping subdomain of the iteration, while retaining the variable parameter cycle. It will be shown that a greater acceleration of the iteration can be obtained by combining overlap and the parameter cycle than by the separate use of either.