A posteriori error control and adaptivity of $hp$-finite elements for mixed and mixed-hybrid methods

Abstract

In this paper, mixed and mixed-hybrid methods for $h$- and $hp$-adaptive finite elements on quadrilateral meshes are discussed for variational equations and, in particular, for variational inequalities. The main result is the derivation of reliable error estimates for mixed methods for the obstacle problem. The estimates rely on the use of a post-processing of the potential in $H^1$ and on the introduction of a certain Lagrange multiplier which is associated with the obstacle constraints. The error estimates consist of the dual norm of the residual, which is defined by an appropriate approximation of the Lagrange multiplier, plus some computable remainder terms. In numerical experiments, the applicability of the post-processing procedure on quadrilateral meshes with multilevel hanging-nodes is verified and the use of the estimates in $h$- and $hp$-adaptive schemes is demonstrated by means of convergence rates and effectivity indices.