Trefftz Discontinuous Galerkin discretization for the Stokes problem

TL;DR

This paper presents a novel Trefftz-DG discretization for the Stokes problem that produces divergence-free solutions, reduces degrees of freedom, and easily incorporates inhomogeneous forces, supported by theoretical analysis and numerical tests.

Contribution

It introduces a new Trefftz-DG method for Stokes equations that achieves divergence-free solutions with fewer degrees of freedom and handles inhomogeneous forces easily.

Findings

Significant reduction in degrees of freedom for higher order polynomials.

Element-wise divergence-free solutions achieved.

Numerical results confirm theoretical error estimates.

Abstract

We introduce a new discretization based on the Trefftz-DG method for solving the Stokes equations. Discrete solutions of a corresponding method fulfill the Stokes equation pointwise within each element and yield element-wise divergence-free solutions. Compared to standard DG methods, a strong reduction of the degrees of freedom is achieved, especially for higher order polynomial degrees. In addition, in contrast to many other Trefftz-DG methods, our approach allows to easily incorporate inhomogeneous right hand sides (driving forces) by using the concept of the embedded Trefftz-DG method. On top of a detailed a priori error analysis, we further compare our approach to standard discontinuous Galerkin Stokes discretizations and present numerical examples.