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DGFEMs for Pure Diffusion Problems

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hp-Version Discontinuous Galerkin Methods on Polygonal and Polyhedral Meshes

Part of the book series: SpringerBriefs in Mathematics ((BRIEFSMATH))

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Abstract

In this chapter we study the stability and hp-version a priori error analysis of the discontinuous Galerkin finite element discretization of a pure diffusion problem. In particular, we develop the underlying theory for two different sets of shape assumptions which the polytopic elements forming the computational mesh must satisfy. In the first instance, we assume that the number of faces each element possesses remains uniformly bounded under mesh refinement, but without a restriction concerning shape-regularity. Secondly, we pursue the analysis in the case when this assumption is violated, i.e., when polytopic elements are permitted to have an arbitrary number of faces under mesh refinement; however, in this setting, a generalized shape-regularity assumption must be satisfied. The relationship between these different mesh assumptions is discussed in detail; indeed, the combination of these conditions allows for very general polytopic meshes to be admitted within our analysis.

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Authors and Affiliations

  1. Department of Mathematics, University of Leicester, Leicester, UK

    Andrea Cangiani, Zhaonan Dong & Emmanuil H. Georgoulis

  2. Department of Mathematics, National Technical University of Athens, Athens, Greece

    Emmanuil H. Georgoulis

  3. School of Mathematical Sciences, University of Nottingham, Nottingham, UK

    Paul Houston

Authors
  1. Andrea Cangiani
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  2. Zhaonan Dong
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  3. Emmanuil H. Georgoulis
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  4. Paul Houston
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Cangiani, A., Dong, Z., Georgoulis, E.H., Houston, P. (2017). DGFEMs for Pure Diffusion Problems. In: hp-Version Discontinuous Galerkin Methods on Polygonal and Polyhedral Meshes. SpringerBriefs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-67673-9_4

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