An Unfitted dG Scheme for Coupled Bulk-Surface PDEs on Complex Geometries
-
Christian Engwer
und
Sebastian Westerheide
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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© 2021 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Sino–German Computational and Applied Mathematics
- An Adaptive Finite Element Scheme for the Hellinger–Reissner Elasticity Mixed Eigenvalue Problem
- Adaptive Directional Compression of High-Frequency Helmholtz Boundary Element Matrices
- Hierarchical Argyris Finite Element Method for Adaptive and Multigrid Algorithms
- On the Threshold Condition for Dörfler Marking
- An Unfitted dG Scheme for Coupled Bulk-Surface PDEs on Complex Geometries
- Dimensionally Consistent Preconditioning for Saddle-Point Problems
- An Optimal Multilevel Method with One Smoothing Step for the Morley Element
- Kernel Embedding Based Variational Approach for Low-Dimensional Approximation of Dynamical Systems
- 𝑯(curl 2)-Conforming Spectral Element Method for Quad-Curl Problems
- Defects in Active Nematics – Algorithms for Identification and Tracking
- Dual-Weighted Residual A Posteriori Error Estimates for a Penalized Phase-Field Slit Discontinuity Problem
- Fast Tensor Product Schwarz Smoothers for High-Order Discontinuous Galerkin Methods
- Error Analysis of an Unconditionally Energy Stable Local Discontinuous Galerkin Scheme for the Cahn–Hilliard Equation with Concentration-Dependent Mobility
Artikel in diesem Heft
- Frontmatter
- Sino–German Computational and Applied Mathematics
- An Adaptive Finite Element Scheme for the Hellinger–Reissner Elasticity Mixed Eigenvalue Problem
- Adaptive Directional Compression of High-Frequency Helmholtz Boundary Element Matrices
- Hierarchical Argyris Finite Element Method for Adaptive and Multigrid Algorithms
- On the Threshold Condition for Dörfler Marking
- An Unfitted dG Scheme for Coupled Bulk-Surface PDEs on Complex Geometries
- Dimensionally Consistent Preconditioning for Saddle-Point Problems
- An Optimal Multilevel Method with One Smoothing Step for the Morley Element
- Kernel Embedding Based Variational Approach for Low-Dimensional Approximation of Dynamical Systems
- 𝑯(curl 2)-Conforming Spectral Element Method for Quad-Curl Problems
- Defects in Active Nematics – Algorithms for Identification and Tracking
- Dual-Weighted Residual A Posteriori Error Estimates for a Penalized Phase-Field Slit Discontinuity Problem
- Fast Tensor Product Schwarz Smoothers for High-Order Discontinuous Galerkin Methods
- Error Analysis of an Unconditionally Energy Stable Local Discontinuous Galerkin Scheme for the Cahn–Hilliard Equation with Concentration-Dependent Mobility
