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/$$hp$$hp discretizations of elliptic boundary-value problems. The main contribution of the paper is a careful construction of a multidimensional Riesz basis in H1, 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

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

VERANI, MARCO
2015-01-01

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/$$hp$$hp discretizations of elliptic boundary-value problems. The main contribution of the paper is a careful construction of a multidimensional Riesz basis in H1, 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
2015
Adaptivity; Convergence; Optimality; Riesz basis; Spectral elements; Software; Computational Theory and Mathematics; Theoretical Computer Science; Engineering (all)
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11311/961904
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