Coupling H(DIV) and H1 Finite Element Approximations for a Poisson Problem

Abstract

The main purpose of this article is to approximate an elliptic problem coupling classical Galerkin and H(div) formulations. As a model problem we consider the Laplace equation on two or three dimensional domain. The
domain is split into two non-overlapping subdomains. On the first one, the problem is approximated using classical Galerkin method. On the other one, the mixed formulation is applied. On the interface, the continuity of flux and pressure is imposed strongly using the transmission condition. The resulting formulation is a saddle point problem which is analysed for stability, existence and uniqueness using Brezzi’s theory.