Unified Interface Flux Evaluation in a General Discontinuous Galerkin Spectral Element Framework

Abstract. High-order discontinuous Galerkin spectral element methods (DGSEM) have received growing attention and development in recent years, especially in the regime of computational fluid dynamics. The inherent flexibility of the DG approach in handling nonconforming interfaces, such as those encountered in moving geometries or [Formula: see text]-refinement, presents a significant advantage for real-world simulations. Despite the well-established mathematical framework of DG methods, practical implementation challenges persist in boosting performance and capability. Most previous studies only focus on certain choices of element shape or basis type in a structured mesh, although they have demonstrated the capability of DGSEM in complex flow simulations. This work discusses low-cost and unified interface flux evaluation approaches for general spectral elements in unstructured meshes, alongside their implementations in the open-source spectral element framework, Nektar++. The initial motivation arises from the discretization of Helmholtz equations by the symmetric interior penalty method, in which the system matrix can easily become nonsymmetric if the flux is not properly evaluated on nonconforming interfaces. We focus on the polynomial nonconforming case in this work, but extending to the geometric nonconforming case is theoretically possible. Comparisons of different approaches, tradeoffs, and performance of our initial matrix-free implementation are also included, contributing to the broader discourse on high-performance spectral element method implementations.