Stabilized CutDG methods for advection-reaction problems

TL;DR

This paper introduces stabilized cut discontinuous Galerkin methods for advection-reaction problems on embedded domains, ensuring robustness and optimal error estimates regardless of how the domain boundary intersects the mesh.

Contribution

The paper develops novel ghost penalty stabilization techniques for CutDG methods, providing robustness and optimal error estimates for advection-reaction problems on unfitted meshes.

Findings

Robustness of methods regardless of cut configurations

Optimal a priori error estimates proven

Numerical experiments confirm theoretical results

Abstract

We develop novel stabilized cut discontinuous Galerkin (CutDG) methods for advection-reaction problems. The domain of interest is embedded into a structured, unfitted background mesh in $\mathbb{R}^d$ where the domain boundary can cut through the mesh in an arbitrary fashion. To cope with robustness problems caused by small cut elements, we introduce ghost penalties in the vicinity of the embedded boundary to stabilize certain (semi)-norms associated with the advection and reaction operator. A few abstract assumptions on the ghost penalties are identified enabling us to derive geometrically robust and optimal a priori error and condition number estimates for the stationary advection-reaction problem which hold irrespective of the particular cut configuration. Possible realizations of suitable ghost penalties are discussed. The theoretical results are corroborated by a number of computational studies for various approximation orders and for two and three-dimensional test problems.