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A dG Approach to Higher Order ALE Formulations in Time

  • Andrea BonitoEmail author
  • Irene Kyza
  • Ricardo H. Nochetto
Chapter
Part of the The IMA Volumes in Mathematics and its Applications book series (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.

Keywords

ALE formulations Moving domains Domain velocity Material derivative Discrete Reynolds’ identities dG-methods in time Stability Geometric conservation law 

Notes

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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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Andrea Bonito
    • 1
    Email author
  • Irene Kyza
    • 2
    • 3
  • Ricardo H. Nochetto
    • 4
  1. 1.Department of MathematicsTexas A&M UniversityCollege StationUSA
  2. 2.Division of MathematicsUniversity of DundeeDundeeScotland, UK
  3. 3.Institute of Applied and Computational Mathematics–FORTHHeraklion-CreteGreece
  4. 4.Department of Mathematics and Institute of Physical Science and TechnologyUniversity of MarylandCollege ParkUSA

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