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

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Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 114))

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.

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Acknowledgements

We would like to thank Prof. El Haj and Dr. Kuiper for supplying the data needed to generate the scaffold geometry shown in Figs. 1 and 5. Paola Antonietti has been partially funded by SIR Project n. RBSI14VT0S PolyPDEs: Non-conforming polyhedral finite element methods for the approximation of partial differential equations funded by MIUR and by the Indam-GNCS Project 2015: Non-standard numerical methods for Geophysics. Joe Collis acknowledges the financial support of the EPSRC under the grant EP/K039342/1. Andrea Cangiani was partially supported by the EPSRC under the grant EP/L022745/1.

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Authors and Affiliations

  1. MOX, Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133, Milano, Italy

    Paola F. Antonietti

  2. Department of Mathematics, University of Leicester, Leicester, LE1 7RH, UK

    Andrea Cangiani, Zhaonan Dong & Emmanuil H. Georgoulis

  3. School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, UK

    Joe Collis & Paul Houston

  4. School of Applied Mathematical and Physical Sciences, National Technical University of Athens, Zografou, 15780, Greece

    Emmanuil H. Georgoulis

  5. School of Engineering and Computing Sciences, Durham University, South Road, Durham, DH1 3LE, UK

    Stefano Giani

Authors
  1. Paola F. Antonietti
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  2. Andrea Cangiani
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  3. Joe Collis
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  4. Zhaonan Dong
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  5. Emmanuil H. Georgoulis
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  6. Stefano Giani
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  7. Paul Houston
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Corresponding author

Correspondence to Paul Houston .

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Editors and Affiliations

  1. Department of Mathematics & Statistics, University of Strathclyde, Glasgow, United Kingdom

    Gabriel R. Barrenechea

  2. Istituto di Matematica Applicata e Tecnologie Informatiche - Pavia, Consiglio Nazionale delle Ricerche, Pavia, Italy

    Franco Brezzi

  3. Department of Mathematics, University of Leicester, Leicester, United Kingdom

    Andrea Cangiani

  4. Department of Mathematics, National Technical University of Athens, Zografou, Greece

    Emmanuil H. Georgoulis

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Antonietti, P.F. et al. (2016). Review of Discontinuous Galerkin Finite Element Methods for Partial Differential Equations on Complicated Domains. In: Barrenechea, G., Brezzi, F., Cangiani, A., Georgoulis, E. (eds) Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations. Lecture Notes in Computational Science and Engineering, vol 114. Springer, Cham. https://doi.org/10.1007/978-3-319-41640-3_9

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