On the stability of approximations for the Stokes problem using different finite element spaces for each component of the velocity

TL;DR

This paper investigates the stability of mixed finite element approximations for the Stokes problem when different FE spaces are used for each velocity component, introducing new combinations and stability analysis techniques.

Contribution

It introduces new FE space combinations for velocity components, analyzes their stability on unstructured meshes, and proposes a post-processing method to ensure mesh uniformity.

Findings

Certain FE combinations are stable on unstructured meshes.

A post-processing technique converts any mesh into an unstructured mesh.

Numerical simulations confirm the theoretical stability results.

Abstract

This paper studies the stability of velocity-pressure mixed approximations of the Stokes problem when different finite element (FE) spaces for each component of the velocity field are considered. We consider some new combinations of continuous FE reducing the number of degrees of freedom in some velocity components. Although the resulting FE combinations are not stable in general, by using the Stenberg's macro-element technique, we show their stability in a wide family of meshes (namely, in uniformly unstructured meshes). Moreover, a post-processing is given in order to convert any mesh family in an uniformly unstructured mesh family. Finally, some 2D and 3D numerical simulations are provided agree with the previous analysis.