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

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.