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A dG Approach to Higher Order ALE Formulations in Time

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Recent Developments in Discontinuous Galerkin Finite Element Methods for Partial Differential Equations

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 157))

Abstract

We review recent results (Bonito et al., SIAM J. Numer. Anal., to appear; Bonito et al., Numer. Math., to appear; Bonito et al., in preparation) on time-discrete discontinuous Galerkin (dG) methods for advection-diffusion model problems defined on deformable domains and written on the arbitrary Lagrangian Eulerian (ALE) framework. ALE formulations deal with PDEs on deformable domains upon extending the domain velocity from the boundary into the bulk with the purpose of keeping mesh regularity. We describe the construction of higher order in time numerical schemes enjoying stability properties independent of the arbitrary extension chosen. Our approach is based on the validity of Reynolds’ identity for dG methods which generalize to higher order schemes the geometric conservation law (GCL) condition. Stability, a priori and a posteriori error analyses are briefly discussed and illustrated by insightful numerical experiments.

AMS(MOS) subject classifications. 65M12, 65M15, 65M50, 65M60.

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Acknowledgements

The work of the Andrea Bonito author was supported in part by NSF Grant DMS-0914977. The work of the Irene Kyza author was supported in part by the European Social Fund (ESF)-European Union (EU) and National Resources of the Greek State within the framework of the Action “Supporting Postdoctoral Researchers” of the Operational Programme “Education and Lifelong Learning (EdLL)”. The work of the Ricardo H. Nochetto author was supported in part by NSF Grants DMS-0807811 and DMS-1109325.

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Authors and Affiliations

  1. Department of Mathematics, Texas A&M University, College Station, TX, 77843-3368, USA

    Andrea Bonito

  2. Division of Mathematics, University of Dundee, Dundee, DD1 4HN, Scotland, UK

    Irene Kyza

  3. Institute of Applied and Computational Mathematics–FORTH, Nikolaou Plastira 100, Vassilika Vouton, Heraklion-Crete, Greece

    Irene Kyza

  4. Department of Mathematics and Institute of Physical Science and Technology, University of Maryland, College Park, MD, 20742-4015, USA

    Ricardo H. Nochetto

Authors
  1. Andrea Bonito
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  2. Irene Kyza
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  3. Ricardo H. Nochetto
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Correspondence to Andrea Bonito .

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Editors and Affiliations

  1. Department of Mathematics, The University of Tennessee, Knoxville, Tennessee, USA

    Xiaobing Feng

  2. Department of Mathematics, The University of Tennessee, Knoxville, Tennessee, USA

    Ohannes Karakashian

  3. Department of Mathematics, The University of Tennessee, Knoxville, Tennessee, USA

    Yulong Xing

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Bonito, A., Kyza, I., Nochetto, R.H. (2014). A dG Approach to Higher Order ALE Formulations in Time. In: Feng, X., Karakashian, O., Xing, Y. (eds) Recent Developments in Discontinuous Galerkin Finite Element Methods for Partial Differential Equations. The IMA Volumes in Mathematics and its Applications, vol 157. Springer, Cham. https://doi.org/10.1007/978-3-319-01818-8_10

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