The Crouzeix-Raviart Element for non-conforming dual mixed methods: A Priori Analysis

TL;DR

This paper provides an a priori error analysis of a discontinuous Galerkin scheme using the Crouzeix-Raviart element for dual mixed formulations of Poisson and Stokes problems, demonstrating convergence and optimal error estimates.

Contribution

It introduces a novel application of the Crouzeix-Raviart element in dual mixed methods with a rigorous error analysis and numerical validation.

Findings

Proves well-posedness and convergence of the method.

Establishes optimal local divergence error estimates.

Numerical experiments confirm optimal convergence even for low regularity solutions.

Abstract

Under some regularity assumptions, we report an a priori error analysis of a dG scheme for the Poisson and Stokes flow problem in their dual mixed formulation. Both formulations satisfy a Babu\v{s}ka-Brezzi type condition within the space H(div) x L2. It is well known that the lowest order Crouzeix-Raviart element paired with piecewise constants satisfies such a condition on (broken) H1 x L2 spaces. In the present article, we use this pair. The continuity of the normal component is weakly imposed by penalizing jumps of the broken H(div) component. For the resulting methods, we prove well-posedness and convergence with constants independent of data and mesh size. We report error estimates in the methods natural norms and optimal local error estimates for the divergence error. In fact, our finite element solution shares for each triangle one DOF with the CR interpolant and the divergence is locally the best-approximation for any regularity. Numerical experiments support the findings and suggest that the other errors converge optimally even for the lowest regularity solutions and a crack-problem, as long as the crack is resolved by the mesh.