Analysis of the SBP-SAT Stabilization for Finite Element Methods Part I: Linear Problems

In the hyperbolic community, discontinuous Galerkin (DG) approaches are mainly applied when finite element methods are considered. As the name suggested, the DG framework allows a discontinuity at the element interfaces, which seems for many researchers a favorable property in case of hyperbolic bal...

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Veröffentlicht in:Journal of scientific computing Jg. 85; H. 2; S. 43
Hauptverfasser: Abgrall, R., Nordström, J., Öffner, P., Tokareva, S.
Format: Journal Article
Sprache:Englisch
Veröffentlicht: New York Springer US 01.11.2020
Springer Nature B.V
Springer
Schlagworte:
Algorithms
Approximation
Boundary conditions
Computational Mathematics and Numerical Analysis
Continuous Galerkin
Differential equations
Discontinuity
Finite element method
Galerkin method
Hyperbolic conservation laws
Initial-boundary value problem
Mathematical analysis
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematics
MATHEMATICS AND COMPUTING
Mathematics and Statistics
Methods
Operators (mathematics)
Perception
Polynomials
Quadratures
Simultaneous approximation terms
Stability
Stabilization
Theoretical
Unstructured grids (mathematics)
Hyperbolic conservation laws
Continuous Galerkin
Initial-boundary value problem
Stability
Simultaneous approximation terms
ISSN:0885-7474, 1573-7691, 1573-7691
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Zusammenfassung:In the hyperbolic community, discontinuous Galerkin (DG) approaches are mainly applied when finite element methods are considered. As the name suggested, the DG framework allows a discontinuity at the element interfaces, which seems for many researchers a favorable property in case of hyperbolic balance laws. On the contrary, continuous Galerkin methods appear to be unsuitable for hyperbolic problems and there exists still the perception that continuous Galerkin methods are notoriously unstable. To remedy this issue, stabilization terms are usually added and various formulations can be found in the literature. However, this perception is not true and the stabilization terms are unnecessary, in general. In this paper, we deal with this problem, but present a different approach. We use the boundary conditions to stabilize the scheme following a procedure that are frequently used in the finite difference community. Here, the main idea is to impose the boundary conditions weakly and specific boundary operators are constructed such that they guarantee stability. This approach has already been used in the discontinuous Galerkin framework, but here we apply it with a continuous Galerkin scheme. No internal dissipation is needed even if unstructured grids are used. Further, we point out that we do not need exact integration, it suffices if the quadrature rule and the norm in the differential operator are the same, such that the summation-by-parts property is fulfilled meaning that a discrete Gauss Theorem is valid. This contradicts the perception in the hyperbolic community that stability issues for pure Galerkin scheme exist. In numerical simulations, we verify our theoretical analysis.
Bibliographie:ObjectType-Article-1
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89233218CNA000001; 200021_175784; FK-19-104; 200021_153604
USDOE
UZH
SNF
LA-UR-19-32410
ISSN:0885-7474
1573-7691
1573-7691
DOI:10.1007/s10915-020-01349-z