Analysis of a hybridized/interface stabilized finite element method for the Stokes equations

TL;DR

This paper presents a stability and error analysis of a hybridized discontinuous Galerkin finite element method for Stokes equations, highlighting its local conservation, stability, and simplicity compared to standard methods.

Contribution

It introduces a stable, locally conservative hybridized finite element method for Stokes equations with a conforming Galerkin-like structure, enhancing simplicity and flexibility.

Findings

The method is inf-sup stable for various spaces.

It achieves  error estimates.

It maintains a conforming Galerkin algebraic structure.

Abstract

Stability and error analysis of a hybridized discontinuous Galerkin finite element method for Stokes equations is presented. The method is locally conservative, and for particular choices of spaces the velocity field is point-wise solenoidal. It is shown that the method is inf-sup stable for both equal-order and locally Taylor--Hood type spaces, and \emph{a priori} error estimates are developed. The considered method can be constructed to have the same global algebraic structure as a conforming Galerkin method, unlike standard discontinuous Galerkin methods that have greater number of degrees of freedom than conforming Galerkin methods on a given mesh. We assert that this method is amongst the simplest and most flexible finite element approaches for Stokes flow that provide local mass conservation. With this contribution the mathematical basis is established, and this supports the performance of the method that has been observed experimentally in other works.