Review of Discontinuous Galerkin Finite Element Methods for Partial Differential Equations on Complicated Domains

Antonietti, Paola F.; Cangiani, Andrea; Collis, Joe; Dong, Zhaonan; Georgoulis, Emmanuil H.; Giani, Stefano; Houston, Paul

Abstract

The numerical approximation of partial differential equations (PDEs) posed on complicated geometries, which include a large number of small geometrical features or microstructures, represents a challenging computational problem. Indeed, the use of standard mesh generators, employing simplices or tensor product elements, for example, naturally leads to very fine finite element meshes, and hence the computational effort required to numerically approximate the underlying PDE problem may be prohibitively expensive. As an alternative approach, in this article we present a review of composite/agglomerated discontinuous Galerkin finite element methods (DGFEMs) which employ general polytopic elements. Here, the elements are typically constructed as the union of standard element shapes; in this way, the minimal dimension of the underlying composite finite element space is independent of the number of geometrical features. In particular, we provide an overview of hp-version inverse estimates and approximation results for general polytopic elements, which are sharp with respect to element facet degeneration. On the basis of these results, a priori error bounds for the hp-DGFEM approximation of both second-order elliptic and first-order hyperbolic PDEs will be derived. Finally, we present numerical experiments which highlight the practical application of DGFEMs on meshes consisting of general polytopic elements.

Citation

Antonietti, P. F., Cangiani, A., Collis, J., Dong, Z., Georgoulis, E. H., Giani, S., & Houston, P. (2016). Review of Discontinuous Galerkin Finite Element Methods for Partial Differential Equations on Complicated Domains. In G. R. Barrenechea, F. Brezzi, A. Cangiani, & E. H. Georgoulis (Eds.), Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations (281-310). Cham: Springer Publishing Company. https://doi.org/10.1007/978-3-319-41640-3_9

• Acceptance Date: Apr 9, 2016
• Online Publication Date: Oct 4, 2016
• Publication Date: Nov 22, 2016
• Deposit Date: Aug 27, 2015
• Publicly Available Date: Oct 5, 2017
• Publisher: Springer Publishing Company
• Peer Reviewed: Not Peer Reviewed
• Pages: 281-310
• Series Title: Lecture Notes in Computational Science and Engineering
• Series Number: 114
• Series ISSN: 1439-7358
• Book Title: Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations
• Chapter Number: 9
• ISBN: 9783319416380
• DOI: https://doi.org/10.1007/978-3-319-41640-3_9
• Public URL: https://nottingham-repository.worktribe.com/output/987662
• Publisher URL: https://link.springer.com/chapter/10.1007%2F978-3-319-41640-3_9