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

Abstract

In the hyperbolic community, discontinuous Galerkin approaches are mainly applied when finite element methods are considered. As the name suggested, the DG framework allows a discontinuity at the element interfaces, which seems for many researchers a favorable property in case of hyperbolic balance laws. On the contrary, continuous Galerkin method obtained from a straightforward discretisation of the weak form of the PDEs appear to be unsuitable for hyperbolic problems. To remedy this issue, stabilization terms are usually added and various formulations can be found in the literature. There exists still the perception that continuous Galerkin methods are not suited to hyperbolic problems, and the reason of this is the continuity of the approximation. However, this perception is not true and the stabilization terms can be removed, in general, provided the boundary conditions are suitable. In this paper, we deal with this problem, and present a different approach. We use the boundary conditions to stabilize the scheme following a procedure that are frequently used in the finite difference community. Here, the main idea is to impose the boundary conditions weakly and specific boundary operators are constructed such that they guarantee stability. This approach has already been used in the DG framework, but here we apply it with a continuous Galerkin scheme. No internal dissipation is needed even if unstructured grids are used. Further, we point out that we do not need exact integration, it suffices if the quadrature rule and the norm in the differential operator are the same, such that the summation-by-parts (SBP) property is fulfilled meaning that a discrete Gauss Th. is valid. This contradicts the perception in the hyperbolic community that stability issues for pure Galerkin scheme exist. In numerical simulations, we verify our theoretical analysis.