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.
%0 Journal Article
%1 stoter2022discontinuous
%A Stoter, Stein K.F.
%A Cockburn, Bernardo
%A Hughes, Thomas J.R.
%A Schillinger, Dominik
%D 2022
%J Computer Methods in Applied Mechanics and Engineering
%K 65m60-pdes-ibvps-finite-elements 65n30-pdes-bvps-finite-elements discontinuous-galerkin
%P 114220
%R https://doi.org/10.1016/j.cma.2021.114220
%T Discontinuous Galerkin methods through the lens of variational multiscale analysis
%U https://www.sciencedirect.com/science/article/pii/S004578252100551X
%V 388
%X 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.
@article{stoter2022discontinuous,
abstract = {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.},
added-at = {2022-09-28T00:47:53.000+0200},
author = {Stoter, Stein K.F. and Cockburn, Bernardo and Hughes, Thomas J.R. and Schillinger, Dominik},
biburl = {https://www.bibsonomy.org/bibtex/22459ff0afe3b38db1536a9149277ff08/gdmcbain},
doi = {https://doi.org/10.1016/j.cma.2021.114220},
interhash = {a88cfa8e31dc67b6e5291a4431c3774a},
intrahash = {2459ff0afe3b38db1536a9149277ff08},
issn = {0045-7825},
journal = {Computer Methods in Applied Mechanics and Engineering},
keywords = {65m60-pdes-ibvps-finite-elements 65n30-pdes-bvps-finite-elements discontinuous-galerkin},
pages = 114220,
timestamp = {2022-09-28T00:47:53.000+0200},
title = {Discontinuous Galerkin methods through the lens of variational multiscale analysis},
url = {https://www.sciencedirect.com/science/article/pii/S004578252100551X},
volume = 388,
year = 2022
}