Dual-primal mixed finite elements for elliptic problems

Abstract

In this article, a novel dual-primal mixed formulation for second-order elliptic problems is proposed and analyzed. The Poisson model problem is considered for simplicity. The method is a Petrov—Galerkin mixed formulation, which arises from the one-element formulation of the problem and uses trial functions less regular than the test functions. Thus, the trial functions need not be continuous while the test functions must satisfy some regularity constraint. Existence and uniqueness of the solution are proved by using the abstract theory of Nicolaides for generalized saddle-point problems. The Helmholtz Decomposition Principle is used to prove the inf-sup conditions in both the continuous and the discrete cases. We propose a family of finite elements valid for any order, which employs piecewise polynomials and Raviart—Thomas elements. We show how the method, with this particular choice of the approximation spaces, is linked to the superposition principle, which holds for linear problems and to the standard primal and dual formulations, addressing how this can be employed for the solution of the final linear system. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 137–151, 2001