Variational multiscale stabilization of high-order spectral elements for the advection-diffusion equation

One major issue in the accurate solution of advection-dominated problems by means of high-order methods is the ability of the solver to maintain monotonicity. This problem is critical for spectral elements, where Gibbs oscillations may pollute the solution. However, typical filter-based stabilization techniques used with spectral elements are not monotone. In this paper, residual-based stabilization methods originally derived for finite elements are constructed and applied to high-order spectral elements. In particular, we show that the use of the variational multiscale (VMS) method greatly improves the solution of the transport-diffusion equation by reducing over- and under-shoots, and can be therefore considered an alternative to filter-based schemes. We also combine these methods with discontinuity capturing schemes (DC) to suppress oscillations that may occur in proximity of boundaries or internal layers. Additional improvement in the solution is also obtained when a method that we call FOS (for First-Order Subcells) is used in combination with VMS and DC. In the regions where discontinuities occur, FOS subdivides a spectral element of order p into p² subcells and then uses 1st-order basis functions and integration rules on every subcell of the element. The algorithms are assessed with the solution of classical steady and transient 1D, 2D, and pseudo-3D problems using spectral elements up to order 16.