Dual-primal mixed finite elements for elliptic problems

in this article, a novel dual-primal mixed formulation for second-order ell iptic problems is proposed and analyzed. The Poisson model problem is consi dered for simplicity. The method is a Petrov-Galerkin mixed formulation, wh ich arises from the one-element formulation of the problem and uses trial f unctions less regular than the test functions. Thus, the trial functions ne ed not be continuous while the test functions must satisfy some regularity constraint. Existence and uniqueness of the solution are proved by using th e abstract theory of Nicolaides for generalized saddle-point problems. The Helmholtz Decomposition Principle is used to prove the inf-sup conditions i n both the continuous and the discrete cases. We propose a family of finite elements valid for any order, which employs piecewise polynomials and Ravi art-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 s ystem. (C) 2001 John Wiley & Sons. Inc.