Abstract
The unfitted discontinuous Galerkin (UDG) method allows for conservative dG discretizations of partial differential equations (PDEs) based on cut cell meshes. It is hence particularly suitable for solving continuity equations on complex-shaped bulk domains.
In this paper based on and extending the PhD thesis of the second author, we show how the method can be transferred to PDEs on curved surfaces. Motivated by a class of biological model problems comprising continuity equations on a static bulk domain and its surface, we propose a new UDG scheme for bulk-surface models.
The method
combines ideas of extending surface PDEs to
higher-dimensional bulk domains with concepts of trace finite
element methods.
A particular focus is given to the necessary steps to retain
discrete analogues to conservation laws of the discretized PDEs.
A high degree
of geometric flexibility is achieved by using a level set
representation of the geometry. We present theoretical
results to prove stability of the method and to investigate its
conservation properties. Convergence is shown in an
energy norm and numerical results show optimal convergence order in
bulk/surface
Funding source: Deutsche Forschungsgemeinschaft
Award Identifier / Grant number: 2044-390685587
Award Identifier / Grant number: EN 1042/4-1
Funding statement: Christian Engwer was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy EXC 2044-390685587, Mathematics Münster: Dynamics–Geometry–Structure. Sebastian Westerheide was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), grant no. EN 1042/4-1: “Massenerhaltende Kopplung von Oberflächen- und Volumenprozessen auf impliziten, zeitabhängigen Gebieten”.
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