A dynamic high-order variational multiscale method on unstructured meshes for transport problems

Abstract

It is well known that the Galerkin method yields a solution with numerical/spurious oscillations in advection-dominated transport problems.Specifically when the cell Peclet or Reynolds number is high with an under resolution of the solution, e.g., in case of an under-resolved boundary layer solution. A popular technique to resolve this issue is to use a stabilization term, in particular one employing the variational multiscale (VMS) and related stabilized methods. The focus of this work is the formulation and application of a dynamic high-order VMS approach on unstructured meshes for stationary and transient transport problems governed by partial differential equations. The current dynamic procedure takes in the given structure/form of the stabilization parameter with unknown coefficients and computes them dynamically in a local fashion resulting in a dynamic VMS-based finite element method. A variational Germano identity (VGI) based local procedure suitable for unstructured meshes and high orders is developed to perform the dynamic computation of the coefficients in the stabilization parameter in a local fashion.The overall dynamic procedure based on the local VGI (LVGI) relies on a sequence of locally coarsened spaces, i.e., secondary coarse-scale spaces, that are constructed from the primary coarse-scale space. To make the current procedure practical, any locally coarser solution is reconstructed from the primary coarse-scale solution, which is done over local patches. Further, averaging steps are employed to make the local dynamic procedure robust. A wide range of stationary and transient problems are considered to demonstrate the suitability of the current high-order LVGI-based dynamic procedure together with different subscale models. Both uniform and nonuniform meshes are employed. Different \timestep sizes are used to evaluate the behavior of different subscale models and stabilization parameters with small and large \timestep sizes. Orders up to $p=7$ are considered. In summary, the current dynamic high-order VMS method is shown to be effective and provide more accurate results (especially on a coarse discretization) for both stationary and transient problems.