An Enriched Galerkin Method for the Stokes Equations

TL;DR

This paper introduces an enriched Galerkin method for the Stokes equations that reduces degrees of freedom and maintains optimal convergence, with scalable solvers validated through numerical experiments.

Contribution

The paper proposes a novel enriched Galerkin scheme combining linear velocity and constant pressure elements, improving efficiency and solver scalability for Stokes problems.

Findings

Reduced degrees of freedom compared to standard methods

Optimal convergence of the scheme

Robust, scalable linear solvers demonstrated

Abstract

We present a new enriched Galerkin (EG) scheme for the Stokes equations based on piecewise linear elements for the velocity unknowns and piecewise constant elements for the pressure. The proposed EG method augments the conforming piecewise linear space for velocity by adding an additional degree of freedom which corresponds to one discontinuous linear basis function per element. Thus, the total number of degrees of freedom is significantly reduced in comparison with standard conforming, non-conforming, and discontinuous Galerkin schemes for the Stokes equation. We show the well-posedness of the new EG approach and prove that the scheme converges optimally. For the solution of the resulting large-scale indefinite linear systems we propose robust block preconditioners, yielding scalable results independent of the discretization and physical parameters. Numerical results confirm the convergence rates of the discretization and also the robustness of the linear solvers for a variety of test problems.