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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 and Houston, Paul 2016. Review of discontinuous Galerkin finite element methods for partial differential equations on complicated domains. Presented at: Building bridges: connections and challenges in modern approaches to numerical partial differential equations, Durham, 8-16 July 2014. Published in: Barrenechea, Gabriel R., Brezzi, Franco, Cangiani, Andrea and Georgoulis, Emmanuil H. eds. Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations. Lecture Notes in Computational Science and Engineering Cham: Springer, pp. 279-308. 10.1007/978-3-319-41640-3_9

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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.

Item Type: Conference or Workshop Item (Paper)
Date Type: Publication
Status: Published
Schools: Mathematics
Publisher: Springer
ISBN: 978-3-319-41640-3
ISSN: 1439-7358
Last Modified: 06 Feb 2020 15:20

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