Quasi oscillation-free high order spectral elements using a numerical subgrid approach

A major issue in the accurate numerical approximation of transport dominated problems
by high-order methods is the ability of the solver to limit the spurious oscillations that affect the solution. Classical filter-based stabilization used with spectral elements may suffer in this respect, hence suggesting the search for suitable alternatives. In this talk, we will show how the Variational Multiscale Stabilization (VMS) method originally derived for finite elements is built for high-order spectral elements. We will demonstrate how VMS greatly improves the solution of the transport-diffusion equation by reducing over- and under-shoots while preserving spectral accuracy, and can therefore be considered an alternative to filter-based schemes. To further suppress oscillations that may occur in the proximity of internal and boundary layers, VMS is coupled to a properly designed discontinuity capturing scheme. Results are shown for 2D, and pseudo-3D problems using spectral elements up to order 16.