Contraction and optimality properties of an adaptive Legendre-Galerkin method: the multi-dimensional case
Keywords
Advanced Numerical Methods for Scientific Computing
Code:
33/2014
Title:
Contraction and optimality properties of an adaptive Legendre-Galerkin method: the multi-dimensional case
Date:
Wednesday 6th August 2014
Author(s):
Canuto, C.; Simoncini, V.; Verani, M.
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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.