Crouzeix-Raviart and Raviart-Thomas finite-element error analysis on anisotropic meshes violating the maximum-angle condition

TL;DR

This paper provides error estimates and establishes equivalence between Crouzeix-Raviart and Raviart-Thomas finite-element methods for the Poisson problem on anisotropic meshes without shape-regularity constraints, supported by numerical validation.

Contribution

It offers new error analysis for these finite-element methods on anisotropic meshes violating common shape conditions and proves their equivalence.

Findings

Error estimates for nonconforming and Raviart-Thomas methods on anisotropic meshes.

Equivalence between Raviart-Thomas and enriched Crouzeix-Raviart methods.

Numerical results confirm theoretical error bounds.

Abstract

We investigate the piecewise linear nonconforming Crouzeix-Raviar and the lowest order Raviart-Thomas finite-element methods for the Poisson problem on three-dimensional anisotropic meshes. We first give error estimates of the Crouzeix-Raviart and the Raviart-Thomas finite-element approximate problems. We next present the equivalence between the Raviart-Thomas finite-element method and the enriched Crouzeix-Raviart finite-element method. We emphasise that we do not impose either shape-regular or maximum-angle condition during mesh partitioning. Numerical results confirm the results that we obtained.