Interior estimates for the wavelet Galerkin method
type: Item
instance of: scholarly article
title: Interior estimates for the wavelet Galerkin method
zbMATH Open document ID: 0888.65113
author: Silvia Bertoluzza
published in: Numerische Mathematik
publication date: 1997-12-10T00:00:00Z
review text: This work is concerned with the use of wavelet bases in determining approximate solutions of partial differential equations. The model problem involves the Poisson equation, which of course is considered in weak or variational form. Solutions are sought in a space \(V_m\) spanned by a wavelet basis. What is of particular interest in this work is the derivation of interior estimates; specifically, suppose that \(\Omega_0\) and \(\Omega_1\) are two subdomains of the domain \(\Omega\) on which the problem is defined, with the property that \(\Omega_0\subset\subset\Omega_1\subset\subset\Omega\). Then for a general class of discretization spaces it has been shown by \textit{J. A. Nitsche} and \textit{A. H. Schatz} [Math. Comput. 28, 937-958 (1974; Zbl 0298.65071)] that the error in the domain \(\Omega_0\) can be bounded according to \[ |u-u_h|_{H^1(\Omega_0)}\leq C(h^{\ell-1}|u|_{H^\ell(\Omega_1)}+|u-u_h|_{H^{-p}(\Omega)})\tag{1} \] provided that \(u\in H^\ell(\Omega_1)\). Thus a high rate of convergence may be obtained on the domain \(\Omega_0\) if \(u\) is regular on the larger subdomain \(\Omega_1\), and provided that an estimate holds in a negative norm for less smooth solutions. The goal of the authors is to show that the estimate (1) holds also for wavelet bases, under suitable assumptions. The framework is that of biorthogonal wavelets [see, for example \textit{A. Cohen, I. Daubechies} and \textit{J.-C. Feauveau}, Commun. Pure Appl. Math. 45, No. 5, 485-560 (1992; Zbl 0776.42020)]. The authors devote a section to reviewing this family of bases, and setting out the assumptions that the bases are required to satisfy: these include assumptions about the support, and inequalities due to Berstein and Jackson. It is then shown that these assumptions hold for classical biorthogonal bases. Eventually, it is shown that an interior error estimate of type (1) holds under various conditions on the integers \(\ell\) and \(p\). In fact the authors are able to derive more genera…
Mathematics Subject Classification ID: 65N15
Mathematics Subject Classification ID: 65N30
Mathematics Subject Classification ID: 65N12
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 35J05
zbMATH DE Number: 1096030
zbMATH Keywords: wavelet Galerkin method
zbMATH Keywords: error estimates
zbMATH Keywords: elliptic problems
zbMATH Keywords: Poisson equation
zbMATH Keywords: convergence
zbMATH Keywords: Dirichlet problem
MaRDI profile type: Publication
full work available at URL: s002110050301
OpenAlex ID: W2026565352
DOI: 10.1007/S002110050301
description: scientific article; zbMATH DE number 1096030