In this article, we present a theoretical framework for integrating discontinuous Galerkin methods in the variational multiscale paradigm. Our starting point is a projector-based multiscale decomposition of a generic variational formulation that uses broken Sobolev spaces and Lagrange multipliers to accommodate the non-conforming nature at the boundaries of discontinuous Galerkin elements. We show that existing discontinuous Galerkin formulations, including their penalty terms, follow immediately from a specific choice of multiscale projector. We proceed by defining the “fine-scale closure function”, which captures the closure relation between the remaining fine-scale term in the discontinuous Galerkin formulation and the coarse-scale solution via a single integral expression for each basis function in the coarse-scale test space. We show that the projectors that correspond to discontinuous Galerkin methods lead to fine-scale closure functions with more compact support and smaller amplitudes compared to the fine-scale closure function of the classical (conforming) finite element method. This observation provides a new perspective on the natural stability of discontinuous Galerkin methods for hyperbolic problems, and may open the door to rigorously designed variational multiscale based fine-scale models that are suitable for DG methods.
|Original language||English (US)|
|Journal||Computer Methods in Applied Mechanics and Engineering|
|State||Published - Jan 1 2022|
Bibliographical noteFunding Information:
Dominik Schillinger gratefully acknowledges funding from the German Research Foundation through the DFG Emmy Noether Award SCH 1249/2-1.
© 2021 Elsevier B.V.
- Advection–diffusion equation
- Discontinuous Galerkin methods
- Fine-scale closure function
- Fine-scale Green's function
- Local discontinuous Galerkin method
- Variational multiscale method