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



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.


Spectral elements Adaptivity Optimality Convergence Riesz basis 



The authors would like to thank Michele Benzi for helpful discussions and for pointing to [23].


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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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