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

High-order discontinuous Galerkin spectral element methods (DGSEM) have received growing attention and development, especially in the regime of computational fluid dynamics in recent years. The inherent flexibility of the discontinuous Galerkin approach in handling non-conforming interfaces, such as those encountered in moving geometries or hp-refinement, presents a significant advantage for real-world simulations. Despite the well-established mathematical framework of DG methods, practical implementation challenges persist to boost 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 the 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 non-symmetric if the flux is not properly evaluated on non-conforming interfaces. We focus on the polynomial non-conforming case in this work but extending to the geometric non-conforming case is theoretically possible. Comparisons of different approaches, trade-offs, and performance of benchmark of our initial matrix-free implementation are also included, contributing to the broader discourse on high-performance spectral element method implementations.