The variational multiscale method for discontinuous Galerkin type finite element formulations

Abstract

In this dissertation, we develop the foundation for a framework that unifies variational multiscale analysis and discontinuous Galerkin type methods. We adopt the variational multiscale principles while using finite element approximation spaces that are flexible on Dirichlet boundaries, or even allowed to be completely discontinuous from element to element. We show that many classical methods follow as particular choices of scale decomposition in the variational multiscale paradigm. The methods that we derive as such are Nitsche's method for the weak enforcement of essential boundary conditions, Bassi-Rebay's first method, the Interior Penalty method, the Local Discontinuous Galerkin method and the Hybridizable Discontinuous Galerkin method. We derive explicitly the projection operators corresponding to each of these formulations. This illustrates that these `non-conformal' methods and the accompanying penalty terms are in complete agreement with the variational multiscale formalism. At the same time, knowledge of these projectors guides our fine-scale modeling efforts. We develop a residual-based model that incorporates the non-vanishing fine scales at the element boundaries. Our model includes additional boundary terms with new model parameters. For all model parameters, we propose a parameter estimation strategy that is effective for both lower- and higher-order basis functions. Our numerical experiments illustrate that the classical residual-based fine-scale model exhibits overly diffusive behavior at boundaries with weakly enforced conditions. The additional terms in the new augmented model counters this defect and improves the quality of the boundary layer approximation.