Coupling Hdiv an H1 Finite Element Approximations for a Poisson Problem

Abstract

The purpose of the paper is to approximate an elliptic problem coupling two different formulations. The domain is split into two non-overlapping sub-domains. On the first one, the problem is approximated using classical Galerkin method where the primal solution p is searched in H ¹ approximation spaces. On the other one, the mixed formulation is applied, which is based on Hdiv and L ² approximation spaces for the dual ∇ p and primal p solutions, respectively. On the interface, the continuity of p and ∇ p is imposed strongly, using transmission conditions. The resulting coupled formulation is a saddle point problem, which is solved for high order hierarchical approximation spaces. Numerical simulations for a test problem show consistent rates of convergence when compared with the corresponding classical and mixed formulations in the whole domain.