New Crouzeix-Raviart elements of even degree: theoretical aspects, numerical performance, and applications to the Stokes' equations

TL;DR

This paper introduces new even degree Crouzeix-Raviart finite element spaces that enable nested bases and variable order methods, with theoretical analysis and numerical validation for Poisson and Stokes' problems.

Contribution

The paper develops novel even degree CR elements with basis functions similar to odd degree cases, facilitating nested bases and variable order discretizations.

Findings

Achieved optimal convergence rates for Poisson problem with smaller stabilization parameters.

Demonstrated effectiveness of new CR elements in Stokes' equations simulations.

Validated theoretical properties through comprehensive numerical experiments.

Abstract

We construct new Crouzeix-Raviart (CR) spaces of even degree $p$ that are spanned by basis functions mimicking those for the odd degree case. Compared to the standard CR gospel, the present construction allows for the use of nested bases of increasing degree and is particularly suited to design variable order CR methods. We analyze a nonconforming discretization of a two dimensional Poisson problem, which requires a DG-type stabilization; the employed stabilization parameter is considerably smaller than that needed in DG methods. Numerical results are presented, which exhibit the expected convergence rates for the $h$-, $p$-, and $hp$-versions of the scheme. We further investigate numerically the behaviour of new even degree CR-type discretizations of the Stokes' equations.