A conforming discontinuous Galerkin finite element method for the Stokes problem on polytopal meshes

TL;DR

This paper introduces a novel conforming discontinuous Galerkin finite element method for the Stokes problem on general meshes, achieving stability without stabilizers and providing optimal error estimates.

Contribution

It develops a conforming DG method for Stokes equations on polytopal meshes that does not require stabilizing terms, unlike most existing methods.

Findings

Achieves optimal-order error estimates in various norms.

Works effectively with polynomial degrees up to four in 2D and 3D.

Validated through numerical examples with low and high order elements.

Abstract

A new discontinuous Galerkin finite element method for the Stokes equations is developed in the primary velocity-pressure formulation. This method employs discontinuous polynomials for both velocity and pressure on general polygonal/polyhedral meshes. Most finite element methods with discontinuous approximation have one or more stabilizing terms for velocity and for pressure to guarantee stability and convergence. This new finite element method has the standard conforming finite element formulation, without any velocity or pressure stabilizers. Optimal-order error estimates are established for the corresponding numerical approximation in various norms. The numerical examples are tested for low and high order elements up to the degree four in 2D and 3D spaces.