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

TL;DR

This paper extends the analysis of SBP-SAT stabilization to nonlinear finite element methods, demonstrating entropy stability and conservation through boundary operator estimation, supported by numerical verification.

Contribution

It introduces a novel approach to ensure entropy stability in nonlinear finite element schemes using boundary operator estimation, building on previous linear stability results.

Findings

Entropy conservation is achieved in nonlinear schemes.

Boundary operator estimation guarantees stability.

Numerical results confirm theoretical predictions.

Abstract

In the 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 linear stable if the boundary conditions are done in the correct way. In this work, we extend this investigation to the non-linear case and focusing on entropy conservation. Switching to entropy variables, we will provide an estimation on the boundary operators also for non-linear problems to guarantee conservation. In numerical simulations, we verify our theoretical analysis.