Iterative solution of a coupled mixed and standard Galerkin discretizationmethod for elliptic problems

In this paper, we consider approximation of a second-order elliptic problem defined on a domain in two-dimensional Euclidean space. Partitioning the d omain into two subdomains, we consider a technique proposed by Wieners and Wohlmuth [9] for coupling mixed finite element approximation on one subdoma in with a standard finite element approximation on the other. In this paper , we study the iterative solution of the resulting linear system of equatio ns. This system is symmetric and indefinite (of saddle-point type). The sta bility estimates for the discretization imply that the algebraic system can be preconditioned by a block diagonal operator involving a preconditioner for H(div) (on the mixed side) and one for the discrete Laplacian (on the f inite element side). Alternatively, we provide iterative techniques based o n domain decomposition. Utilizing subdomain solvers, the composite problem is reduced to a problem defined only on the interface between the two subdo mains. We prove that the interface problem is symmetric, positive definite and well conditioned and hence can be effectively solved by a conjugate gra dient iteration. Copyright (C) 2001 John Wiley & Sons, Ltd.