Iterative Solution of a Combined Mixed and Standard Galerkin Discretization Method 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 domain into two subdomains, we consider a technique proposed by Wieners and Wohlmuth [23] for coupling mixed finite element approximation on one subdomain with a finite element approximation on other. We consider iterative solution of the resulting linear system of equations. This system is symmetric and indefinite (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 Hdiv (on the mixed side) and one for the discrete Laplacian (on the finite element side). Alternatively, we provide iterative techniques based on domain decomposition. Utilizing subdomain solves, the composite problem is reduced to a problem defined only on the interface between the two subdomains. We prove that the interface problem is symmetric, positive definite and well conditioned and hence can be effectively solved by a conjugate gradient iteration.