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Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2014 pp 51–72Cite as

High-Order Adaptive Galerkin Methods

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Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE,volume 106)

Abstract

We design adaptive high-order Galerkin methods for the solution of linear elliptic problems and study their performance. We first consider adaptive Fourier-Galerkin methods and Legendre-Galerkin methods, which offer unlimited approximation power only restricted by solution and data regularity. Their analysis of convergence and optimality properties reveals a sparsity degradation for Gevrey classes. We next turn our attention to the h p-version of the finite element method, design an adaptive scheme which hinges on a recent algorithm by P. Binev for adaptive h p-approximation, and discuss its optimality properties.

Keywords

  • Polynomial Degree
  • Greedy Approach
  • Posteriori Error Estimator
  • Data Oscillation
  • Galerkin Solution

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Acknowledgements

The authors “Claudio Canuto” and “Marco Verani” were partially supported by the Italian national grant PRIN 2012HBLYE4. The author “Ricardo H. Nochetto” was partially supported by NSF grants DMS-1109325 and DMS-1411808.

Author information

Authors and Affiliations

  1. Dipartimento di Scienze Matematiche, Politecnico di Torino, Corso Duca degli Abruzzi 24, I-10129, Torino, Italy

    Claudio Canuto

  2. Department of Mathematics and Institute for Physical Science and Technology, University of Maryland, College Park, MD, USA

    Ricardo H. Nochetto

  3. Korteweg-de Vries Institute for Mathematics, University of Amsterdam, 94248, 1090 GE, Amsterdam, The Netherlands

    Rob Stevenson

  4. MOX-Dipartimento di Matematica, Politecnico di Milano, P.zza Leonardo Da Vinci 32, I-20133, Milano, Italy

    Marco Verani

Authors
  1. Claudio Canuto
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  2. Ricardo H. Nochetto
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  3. Rob Stevenson
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  4. Marco Verani
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Corresponding author

Correspondence to Ricardo H. Nochetto .

Editor information

Editors and Affiliations

  1. School of Computing , University of Utah, Salt Lake City, Utah, USA

    Robert M. Kirby

  2. School of Computing, University of Utah, Salt Lake City, Utah, USA

    Martin Berzins

  3. EPFL-SB-MATHICSE-MCSS, Ecole Polytechnique Fédérale de Lausanne, Lausanne, Switzerland

    Jan S. Hesthaven

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Canuto, C., Nochetto, R.H., Stevenson, R., Verani, M. (2015). High-Order Adaptive Galerkin Methods. In: Kirby, R., Berzins, M., Hesthaven, J. (eds) Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2014. Lecture Notes in Computational Science and Engineering, vol 106. Springer, Cham. https://doi.org/10.1007/978-3-319-19800-2_4

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