Analysis of the SBP-SAT Stabilization for Finite Element Methods Part I: Linear problems
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A pure Galerkin scheme is notoriously unstable. To remedy this issue, stabilization terms are usually added and various formulations can be found in the literature. In this paper, we are also dealing with this problem, but present a different approach. We use the boundary conditions in our investigation in the sense that so called simultaneous approximation terms (SATs) are applied which are frequently used in the finite difference community. Here, the main idea is to impose the boundary conditions weakly. Specific boundary operators are constructed which guarantee stability. The SAT approach has already been used in the discontinuous Galerkin framework, but here we apply it – up to our knowledge – for the first time together with a continuous Galerkin scheme. We demonstrate that a pure continuous Galerkin scheme is stable if the boundary conditions are implemented in the correct way. This contradicts the general perception of stability problems for pure Galerkin schemes. In numerical simulations, we verify our theoretical analysis.
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