Residual-based variational multiscale modeling in a discontinuous Galerkin framework

Abstract

In this work the residual-based variational multiscale method is presented in a discontinuous Galerkin framework. This so-called ‘DG-RVMS’ strategy is developed, extensively verified, and tested on a more complex case.

The proposed DG-RVMS paradigm consists of three principle components. First, the perelement weak form is coupled by manipulation of the fine scale element boundary terms. Next, a fine scale surface model is introduced to make the global coarse scale weak formulation well posed. Finally, the coarse scale jumps and residual can be leveraged to formulate a new volumetric fine scale model. This volumetric fine scale model incorporates the fine scale effects onto the coarse scale solution.

The verification efforts will focus on a number of 1D test cases, concerning linear differential equations. In particular the Poisson equation and an advection-diffusion problem will be investigated. Within a controlled environment the finite element solution can be manipulated at will, by using explicit expressions for the fine scale terms. As an example the H1 and L2 projections of an exact solution are recollected. Each term in the obtained multiscale formulations will be verified by means of these numerical experiments.

Additionally, the multiscale principles will serve to develop fundamentally new insights into the nature of known discontinuous Galerkin formulations. It will be shown that classical formulations, such as the well known interior penalty method, can be interpreted as a specific fine scale model. It will also be shown that upwind numerical fluxes serve as an impromptu solution for the lack of a volumetric fine scale model.

Finally, the DG-RVMS framework will be utilized for a more complex partial differential equation. Thereby its effectiveness as a multiscale model can be assessed. For this purpose the nonlinear transient Burgers equation will be considered. Numerical experiments will consistently show a near order of magnitude decrease in the error in total solution energy. The experiments will make use of discretizations of polynomial order p = 2 to p = 4. The increase of performance is observed for all polynomial orders, and for the complete range of
degrees of freedom in the convergence study.