Stabilized cut discontinuous Galerkin methods for advection-reaction problems on surfaces

TL;DR

This paper introduces a new stabilized cut discontinuous Galerkin method for solving stationary advection-reaction problems on embedded surfaces, ensuring stability and accuracy regardless of how the surface intersects the background mesh.

Contribution

The paper presents a novel CutDG method with stabilization that guarantees stability and error estimates for surface PDEs with arbitrary surface-mesh intersections.

Findings

Method achieves inf-sup stability and optimal error estimates.

Theoretical results are independent of cut configurations.

Numerical examples confirm the theoretical stability and accuracy.

Abstract

We develop a novel cut discontinuous Galerkin (CutDG) method for stationary advection-reaction problems on surfaces embedded in $\mathbb{R}^d$. The CutDG method is based on embedding the surface into a full-dimensional background mesh and using the associated discontinuous piecewise polynomials of order $k$ as test and trial functions. As the surface can cut through the mesh in an arbitrary fashion, we design a suitable stabilization that enables us to establish inf-sup stability, a priori error estimates, and condition number estimates using an augmented streamline-diffusion norm. The resulting CutDG formulation is geometrically robust in the sense that all derived theoretical results hold with constants independent of any particular cut configuration. Numerical examples support our theoretical findings.