Variational Multiscale Stabilization and p-adaptivity of High-Order Spectral Elements for the Convection-Diffusion Equation (Preprint)

摘要

One major issue in the accurate solution of convection-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 the limitations of filter-based schemes. We also combine these methods with discontinuity capturing schemes to suppress oscillations that may occur in proximity of boundary or internal layers. Additional improvement in the solution is also obtained when p-adaptivity is used in combination with VMS in the regions where discontinuities occur. The algorithms are assessed with the solution of classical steady and transient one- and two-dimensional problems using spectral elements up to order 16.