Unfitted Trefftz discontinuous Galerkin methods for elliptic boundary value problems

TL;DR

This paper introduces a new unfitted discontinuous Galerkin method using Trefftz spaces for elliptic boundary value problems, reducing degrees of freedom and computational costs while ensuring stability and accuracy.

Contribution

It develops a unified analysis for geometrically unfitted Trefftz DG methods, including stability, error bounds, and handling small mesh cuts, applicable to various ansatz spaces.

Findings

Reduced degrees of freedom with Trefftz methods

Stable and accurate for unfitted geometries

Numerical results confirm theoretical predictions

Abstract

We propose a new geometrically unfitted finite element method based on discontinuous Trefftz ansatz spaces. Trefftz methods allow for a reduction in the number of degrees of freedom in discontinuous Galerkin methods, thereby, the costs for solving arising linear systems significantly. This work shows that they are also an excellent way to reduce the number of degrees of freedom in an unfitted setting. We present a unified analysis of a class of geometrically unfitted discontinuous Galerkin methods with different stabilisation mechanisms to deal with small cuts between the geometry and the mesh. We cover stability and derive a-priori error bounds, including errors arising from geometry approximation for the class of discretisations for a model Poisson problem in a unified manner. The analysis covers Trefftz and full polynomial ansatz spaces, alike. Numerical examples validate the theoretical findings and demonstrate the potential of the approach.