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

In this paper, we consider approximation of a second-order elliptic problem de ned on a domain in two-dimensional Euclidean space. Partitioning the domain into two subdomains, we consider a technique proposed by Wieners and Wohlmuth [9] for coupling mixed nite element approximation on one subdomain with a standard nite element approximation on the other. In this paper, we study the iterative solution of the resulting linear system of equations. This system is symmetric and inde nite (of saddle-point type). The stability 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 nite element side). Alternatively, we provide iterative techniques based on domain decomposition. Utilizing subdomain solvers, the composite problem is reduced to a problem de ned only on the interface between the two subdomains. We prove that the interface problem is symmetric, positive de nite and well conditioned and hence can be e ectively solved by a conjugate gradient iteration. Copyright ? 2001 John Wiley & Sons, Ltd.