A stabilized cut discontinuous Galerkin framework: I. Elliptic boundary value and interface problems

TL;DR

This paper introduces a stabilized cut discontinuous Galerkin method for solving elliptic boundary and interface problems on complex domains embedded in unfitted meshes, using ghost penalty techniques for robustness.

Contribution

It develops a novel stabilized unfitted DG framework employing ghost penalties, enabling robust error estimates and easy adaptation to complex geometries without cell agglomeration.

Findings

Achieves geometrically robust a priori error estimates.

Demonstrates applicability to high contrast interface problems.

Validates theoretical results with numerical experiments.

Abstract

We develop a stabilized cut discontinuous Galerkin framework for the numerical solution of el- liptic boundary value and interface problems on complicated domains. The domain of interest is embedded in a structured, unfitted background mesh in R d , so that the boundary or interface can cut through it in an arbitrary fashion. The method is based on an unfitted variant of the classical symmetric interior penalty method using piecewise discontinuous polynomials defined on the back- ground mesh. Instead of the cell agglomeration technique commonly used in previously introduced unfitted discontinuous Galerkin methods, we employ and extend ghost penalty techniques from recently developed continuous cut finite element methods, which allows for a minimal extension of existing fitted discontinuous Galerkin software to handle unfitted geometries. Identifying four abstract assumptions on the ghost penalty, we derive geometrically robust a priori error and con- dition number estimates for the Poisson boundary value problem which hold irrespective of the particular cut configuration. Possible realizations of suitable ghost penalties are discussed. We also demonstrate how the framework can be elegantly applied to discretize high contrast interface problems. The theoretical results are illustrated by a number of numerical experiments for various approximation orders and for two and three-dimensional test problems.