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

R. Abgrall, J. Nordström, P. Öffner, S. Tokareva

Research output: Contribution to journalArticlepeer-review

28 Citations (Scopus)

Abstract

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.

Original languageEnglish
Article number43
JournalJournal of Scientific Computing
Volume85
Issue number2
DOIs
Publication statusPublished - 1 Nov 2020

Keywords

  • Continuous Galerkin
  • Hyperbolic conservation laws
  • Initial-boundary value problem
  • Simultaneous approximation terms
  • Stability

ASJC Scopus subject areas

  • Software
  • Theoretical Computer Science
  • Numerical Analysis
  • General Engineering
  • Computational Mathematics
  • Computational Theory and Mathematics
  • Applied Mathematics

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