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. In addition, the nodal assembly leads to a line-based sparsity pattern for first-order operators which can give magnitudes of speed-up and memory reductions compared to traditional stabilized finite element schemes such as the discontinuous Galerkin (DG) method. We present the derivation of the scheme and the analysis of its properties, in particular showing stability using von Neumann analysis. We also show that for 1D constant-coefficient problems, the scheme can be re-written as a version of the Spectral Difference method, which immediately leads to conservation and stability guarantees for any polynomial degrees. We show numerous numerical evidence for its accuracy, efficiency, and high sparsity compared to traditional schemes, on multiple classes of problems including convection-dominated flows, Poisson's equation, and the incompressible Navier-Stokes equations.

翻译：我们提出了一种适用于非结构化四边形网格的高阶精度离散方法。我们的面迎风谱元（FUSE）方法采用与高阶连续Galerkin（CG）方法相同的节点分布，但通过在每个单元内选择特定的节点位置，并在面节点上采用迎风模板。这带来了多项优势，包括减少自由度以及能够直接与CG方法集成。此外，节点组装使得一阶算子产生基于线的稀疏模式，与传统稳定有限元方案（如间断Galerkin（DG）方法）相比，可实现数量级的加速和内存缩减。我们给出了该方案的推导过程及其属性分析，特别地，通过von Neumann分析证明了其稳定性。我们同时证明，对于一维常系数问题，该方案可重新表述为谱差分方法的一种形式，从而立即得到任意多项式阶数下的守恒性与稳定性保证。通过多类问题的数值实验（包括对流主导流动、泊松方程及不可压缩Navier-Stokes方程），我们充分验证了该方法相比传统方案在精度、效率及高稀疏性方面的优势。