A Face-Upwinded Spectral Element Method
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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.
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