Finite element methods for variational problems based on nonconforming dual mixed discretisations

Abstract

A novel hybrid mixed discretisation scheme for the dual formulation of elliptic boundary value problems is presented. It is designed to facilitate the application of reliable a posteriori error estimators based on complementary energy principles. The Lagrange multipliers for the continuity constraints to be imposed on the dual variables are introduced as traces of shape functions which are of Lagrange type. Thus a conforming primal approximation can be obtained by post-processing these multipliers, while the dual approximation need no longer be conforming. The scheme may be viewed as complementing the well known equivalence of conforming dual and nonconforming primal methods. Asymptotic rates of convergence can be established for both dual and primal variables that are of optimal order.