Journal of Scientific Computing

, Volume 63, Issue 3, pp 769–798 | Cite as

Contraction and Optimality Properties of an Adaptive Legendre–Galerkin Method: The Multi-Dimensional Case

Article

Abstract

We analyze the theoretical properties of an adaptive Legendre–Galerkin method in the multidimensional case. After the recent investigations for Fourier–Galerkin methods in a periodic box and for Legendre–Galerkin methods in the one dimensional setting, the present study represents a further step towards a mathematically rigorous understanding of adaptive spectral/\(hp\) discretizations of elliptic boundary-value problems. The main contribution of the paper is a careful construction of a multidimensional Riesz basis in \(H^1\), based on a quasi-orthonormalization procedure. This allows us to design an adaptive algorithm, to prove its convergence by a contraction argument, and to discuss its optimality properties (in the sense of non-linear approximation theory) in certain sparsity classes of Gevrey type.

Keywords

Spectral elements Adaptivity Optimality Convergence Riesz basis 

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Claudio Canuto
    • 1
  • Valeria Simoncini
    • 2
  • Marco Verani
    • 3
  1. 1.Dipartimento di Scienze MatematichePolitecnico di TorinoTurinItaly
  2. 2.Dipartimento di MatematicaUniversità di BolognaBolognaItaly
  3. 3.MOX-Dipartimento di MatematicaPolitecnico di MilanoMilanItaly

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