Stabilized Cut Discontinuous Galerkin Methods for Advection-Reaction Problems
Abstract
We develop novel stabilized cut discontinuous Galerkin 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.
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SIAM Journal on Scientific Computing
Pages: A2620 - A2654
DOI: 10.1137/18M1206461
ISSN (online): 1095-7197
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© 2020, Society for Industrial and Applied Mathematics.
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Submitted: 14 August 2018
Accepted: 28 April 2020
Published online: 9 September 2020
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Kempestiftelserna https://doi.org/10.13039/501100007067 : JCK-1612
Svenska Forskningsrådet Formas https://doi.org/10.13039/501100001862 : 2017-05038
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