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On the Numerical Analysis of Adaptive Spectral/hp Methods for Elliptic Problems

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Book cover Analysis and Numerics of Partial Differential Equations

Part of the book series: Springer INdAM Series ((SINDAMS,volume 4))

Abstract

We provide an overview of the state of the art of adaptive strategies for high-order hp discretizations of partial differential equations; at the same time, we draw attention on some recent results of ours concerning the convergence and complexity analysis of adaptive algorithm of spectral and spectral-element type. Complexity is studied under the assumption that the solution belongs to a sparsity class of exponential type, which means that its best N-term approximation error in the chosen piecewise polynomial basis decays at an exponential rate with respect to N.

The transcendental is not infinite and unattainable tasks, but the neighbor who is within reach in any given situation.

(D. Bonhoeffer)

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Author information

Authors and Affiliations

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

    Claudio Canuto

  2. MOX-Dipartimento di Matematica, Politecnico di Milano, P.zza Leonardo da Vinci 32, 20133, Milan, Italy

    Marco Verani

Authors
  1. Claudio Canuto
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  2. Marco Verani
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Correspondence to Claudio Canuto .

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Editors and Affiliations

  1. Ist. Matematica Applicata e, Tecnologie Informatiche (IMATI), CNR Pavia, Via Ferrata 1, Pavia, 27100, Italy

    Franco Brezzi

  2. Dipartimento di Matematica "F. Casorati", Università degli Studi di Pavia, via Ferrata 1, Pavia, 27100, Pavia, Italy

    Piero Colli Franzone

  3. Dipartimento di Matematica "F. Casorati", Università degli Studi di Pavia, Via Ferrata 1, Pavia, 27100, Italy

    Ugo Gianazza

  4. Dipartimento di Matematica "F. Casorati", Università degli Studi di Pavia, via Ferrata 1, Pavia, 27100, Pavia, Italy

    Gianni Gilardi

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Canuto, C., Verani, M. (2013). On the Numerical Analysis of Adaptive Spectral/hp Methods for Elliptic Problems. In: Brezzi, F., Colli Franzone, P., Gianazza, U., Gilardi, G. (eds) Analysis and Numerics of Partial Differential Equations. Springer INdAM Series, vol 4. Springer, Milano. https://doi.org/10.1007/978-88-470-2592-9_11

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