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Analysis of the SBP-SAT Stabilization for Finite Element Methods Part II: Entropy Stability

Abstract

In the hyperbolic research community, there exists the strong belief that a continuous Galerkin scheme is notoriously unstable and additional stabilization terms have to be added to guarantee stability. In the first part of the series [6], the application of simultaneous approximation terms for linear problems is investigated where the boundary conditions are imposed weakly. By applying this technique, the authors demonstrate that a pure continuous Galerkin scheme is indeed linearly stable if the boundary conditions are imposed in the correct way. In this work, we extend this investigation to the nonlinear case and focus on entropy conservation. By switching to entropy variables, we provide an estimation of the boundary operators also for nonlinear problems, that guarantee conservation. In numerical simulations, we verify our theoretical analysis.

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Notes

  1. 1.

    This term is also referred to as a fluctuation term in literature [8].

  2. 2.

    In the finite difference community \({\underline{\underline{M}}}\) is called norm matrix and is classically abbreviated with P, c.f., [28, 38].

  3. 3.

    In abuse of notation we use \(\mathbf {g}'\) for the derivative of \(\mathbf {g}{\mathbf {n}}\) and do not apply everywhere the normal vector \({\mathbf {n}}\). Similar also for the relation (10).

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Acknowledgements

P.Öffner has been funded by the SNF Grant (Number 200021_175784) and by the UZH Postdoc grant. This research was initiated by a first visit of JN at UZH, and really started during ST postdoc at UZH. This postdoc was funded by an SNF Grant 200021_153604. The Los Alamos unlimited release number is LA-UR-19-32411.

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Correspondence to P. Öffner.

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Abgrall, R., Nordström, J., Öffner, P. et al. Analysis of the SBP-SAT Stabilization for Finite Element Methods Part II: Entropy Stability. Commun. Appl. Math. Comput. (2021). https://doi.org/10.1007/s42967-020-00086-2

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Keywords

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

Mathematics Subject Classification

  • 65M12
  • 65M60
  • 65M70
  • 65M06