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We evaluate the performance of the Community Atmosphere Model's  (CAM) spectral element method on variable-resolution grids using the  shallow-water equations in spherical geometry.  We configure the  method as it is used in CAM, with dissipation of grid scale variance,  implemented using hyperviscosity.  Hyperviscosity is highly scale  selective and grid independent, but does require a resolution-dependent coefficient.  For the spectral element method with  variable-resolution grids and highly distorted elements, we obtain  the best results if we introduce a tensor-based hyperviscosity with  tensor coefficients tied to the eigenvalues of the local element  metric tensor.  The tensor hyperviscosity is constructed so that, for  regions of uniform resolution, it matches the traditional constant-coefficient hyperviscosity.  With the tensor hyperviscosity, the  large-scale solution is almost completely unaffected by the presence  of grid refinement.  This later point is important for climate  applications in which long term climatological averages can be  imprinted by stationary inhomogeneities in the truncation  error. We also evaluate the robustness of the approach with respect  to grid quality by considering unstructured conforming quadrilateral  grids generated with a well-known grid-generating toolkit and grids  generated by SQuadGen, a new open source alternative which produces  lower valence nodes

The spectral element method (SEM) on variable-resolution grids: evaluating grid sensitivity and resolution-aware numerical viscosity