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

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|November 13, 2020
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Summary

Continuous Galerkin methods can be stabilized for hyperbolic problems using boundary conditions, eliminating the need for stabilization terms or internal dissipation. This approach ensures stability even on unstructured grids.

Keywords:
Continuous GalerkinHyperbolic conservation lawsInitial-boundary value problemSimultaneous approximation termsStability

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Area of Science:

  • Computational Mathematics
  • Numerical Analysis
  • Fluid Dynamics

Background:

  • Discontinuous Galerkin (DG) methods are prevalent for hyperbolic problems due to their handling of discontinuities.
  • Continuous Galerkin (CG) methods are often perceived as unstable for hyperbolic equations, necessitating stabilization terms.
  • Existing stabilization techniques for CG methods in hyperbolic systems are often complex and may be unnecessary.

Purpose of the Study:

  • To challenge the perception of CG method instability for hyperbolic problems.
  • To introduce a novel stabilization approach for CG schemes using boundary conditions.
  • To demonstrate the efficacy of CG methods without internal dissipation or complex stabilization.

Main Methods:

  • Weak imposition of boundary conditions using specifically constructed boundary operators.
  • Application of the boundary stabilization technique within a continuous Galerkin framework.
  • Verification of the summation-by-parts property and discrete Gauss Theorem fulfillment.

Main Results:

  • The proposed boundary condition stabilization guarantees stability for CG schemes in hyperbolic problems.
  • Internal dissipation is not required, even with unstructured grids.
  • Exact integration is not necessary; matching quadrature rules and norms suffice for stability.

Conclusions:

  • Continuous Galerkin methods can be effectively stabilized for hyperbolic problems through weak boundary condition imposition.
  • The need for traditional stabilization terms in CG methods for hyperbolic equations is negated.
  • This work provides a stable and efficient alternative to DG methods for hyperbolic balance laws.