An expanded mixed finite element approach via a dual–dual formulation and the minimum residual method

Abstract

We apply an expanded mixed finite element method, which introduces the gradient as a third explicit unknown, to solve a linear second-order elliptic equation in divergence form. Instead of using the standard dual form, we show that the corresponding variational formulation can be written as a dual–dual operator equation. We establish existence and uniqueness of solution for the continuous and discrete formulations, and provide the corresponding error analysis by using Raviart–Thomas elements. In addition, we show that the corresponding dual–dual linear system can be efficiently solved by a preconditioned minimum residual method. Some numerical results, illustrating this fact and the rate of convergence of the mixed finite element method, are also provided.