Skip to main content

Research Repository

Advanced Search

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

Authors

Paola F. Antonietti paola.antonietti@polimi.it

Andrea Cangiani Andrea.Cangiani@le.ac.uk

Joe Collis Joe.Collis@nottingham.ac.uk

Zhaonan Dong zd14@le.ac.uk

Emmanuil H. Georgoulis Emmanuil.Georgoulis@mcs.le.ac.uk

Stefano Giani stefano.giani@durham.ac.uk

PAUL HOUSTON paul.houston@nottingham.ac.uk
Professor of Computational and Applied Maths



Contributors

Gabriel R. Barrenechea
Editor

Franco Brezzi
Editor

Andrea Cangiani
Editor

Emmanuil H. Georgoulis
Editor

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.

Publication Date Nov 22, 2016
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Peer Reviewed Not Peer Reviewed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
APA6 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. Springer Publishing Company. https://doi.org/10.1007/978-3-319-41640-3_9
DOI https://doi.org/10.1007/978-3-319-41640-3_9
Publisher URL https://link.springer.com/chapter/10.1007%2F978-3-319-41640-3_9
Copyright Statement Copyright information regarding this work can be found at the following address: http://eprints.nottingh.../end_user_agreement.pdf

Files

paper_v1.pdf (1.3 Mb)
PDF

Copyright Statement
Copyright information regarding this work can be found at the following address: http://eprints.nottingham.ac.uk/end_user_agreement.pdf





You might also like



Downloadable Citations

;