We present a new high-order accurate discretisation on unstructured meshes of quadrilateral elements. Our Face Upwinded Spectral Element (FUSE) method uses the same node distribution as a high-order continuous Galerkin (CG) method, but with a particular choice of node locations within each element and an upwinded stencil on the face nodes. This results in a number of benefits, including fewer degrees of freedom and straight-forward integration with CG. We present the derivation of the scheme and the analysis of its properties, in particular showing stability using von Neumann analysis. We show numerical evidence for its accuracy and efficiency on multiple classes of problems including convection-dominated flows, Poisson's equation, and the incompressible Navier-Stokes equations.

翻译：我们提出了一种在非结构化四边形网格上的高阶精确离散格式。我们的面迎风谱元法（FUSE）采用了与高阶连续伽辽金（CG）方法相同的节点分布，但在每个单元内特定选择了节点位置，并对面节点采用了迎风格式。这带来了多项优势，包括自由度减少以及与CG方法的直接集成。我们给出了该格式的推导过程及其性质分析，特别通过冯·诺依曼分析证明了其稳定性。我们通过多类问题的数值实验展示了其精度和效率，涵盖对流主导流动、泊松方程以及不可压缩纳维-斯托克斯方程。