Convergence estimates for the wavelet-Galerkin method: Superconvergence at the node points
type: Item
instance of: scholarly article
title: Convergence estimates for the wavelet-Galerkin method: Superconvergence at the node points
zbMATH Open document ID: 0840.65101
DOI: 10.1007/BF03177516
author: Sônia M. Gomes
published in: Advances in Computational Mathematics
publication date: 1996-06-30T00:00:00Z
review text: The author studies approximations of a linear partial differential equation in one space variable and in time, the spatial domain being the real line. The data is assumed to be 1-periodic in space, and Galerkin approximations are sought in finite-dimensional subspaces \(V_h\) of \(L^2(0,1)\) which are spanned by the scaled translates of a basic function \(\phi\). That is, Galerkin approximations are of the form \[ u_h(x,t) = \sum_k v_k(t) \phi(Nx - k) \] where \(N\) is a positive integer and \(h = 1/N\) is the mesh size. The mesh is assumed to be regular, and the coefficients are assumed to be 1-periodic with period \(N\). The problem has been previously studied by (*) \textit{V. Thomée} and \textit{B. Wendroff} [SIAM J. Numer. Anal. 11, 1059-1068 (1974; Zbl 0292.65052)], in which \(\phi\) has been taken to be a B-spline. It is possible to obtain a superconvergence result using this approach. The approach of the paper under review is to extend the study in (*) to the larger space \({\mathcal H}^*_{r,m}\), comprising functions which are \(r\)-regular and whose Fourier transforms are non-zero at the origin, and which have zeros of order \(m\) at the points \(2k\pi\), \(k \neq 0\) (this is the so-called Strang-Fix condition). This space includes for \(m = r\), the B-splines as well as \(r\)-regular multiresolution analyses of \(L^2(\mathbb{R})\). Finally, the superposed asterisk denotes the fact that any member \(\phi \in {\mathcal H}^*_{m,r}\) may be modified to obtain a function \(\phi^*\), also in this space, and with the property that its Fourier transform satisfies \(\phi^*(\xi) = 1+O(\xi^{2m+2})\). The first result obtained is an error estimate for smooth 1-periodic functions. Next, it is shown that the following superconvergence result holds: the \(L^2\)-error in the solution to the initial value problem is of order \(\min(2m + 2 - \mu, m + 1)\) where \(r \geq \mu/2\). In the final section of the paper the results are discussed in the context of wavelet analysis…
Mathematics Subject Classification ID: 65M60
Mathematics Subject Classification ID: 65M15
Mathematics Subject Classification ID: 35K15
Mathematics Subject Classification ID: 65M12
zbMATH DE Number: 826884
zbMATH Keywords: wavelet-Galerkin method
zbMATH Keywords: superconvergence
zbMATH Keywords: Strang-Fix condition
zbMATH Keywords: B-splines
zbMATH Keywords: multiresolution analyses
zbMATH Keywords: error estimate
zbMATH Keywords: wavelet analysis
zbMATH Keywords: scaling functions
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf03177516
OpenAlex ID: W2031186469
description: scientific article; zbMATH DE number 826884