A multiresolution strategy for numerical homogenization
type: Item
instance of: scholarly article
title: A multiresolution strategy for numerical homogenization
zbMATH Open document ID: 0840.65047
author: M. E. Brewster
author: Gregory Beylkin
published in: Applied and Computational Harmonic Analysis
publication date: 1996-06-30T00:00:00Z
review text: The work is concerned with a multiresolution strategy for homogenization of differential equations (ordinary and partial). The authors develop an efficient numerical approach which generates the coefficients of the homogenized equation. The multiresolution analysis (MRA), a notion introduction by \textit{Y. Meyer}, is used [Rev. Mat. Iberoam. 8, No. 2, 115-133 (1991; Zbl 0753.42015)]. The equations (differential or integral) considered in this article can be written in the general form (1) \(Bx + q + \lambda = K(Ax + p)\) where \(A\), \(B\), \(K\) are operators on functions in \(L_2(0,1)\) with values in a given Hilbert space \(H\), \(p\), \(q\), \(x\) are square-integrable functions defined on \([0,1]\) with values in \(H\) and \(\lambda\) is a parameter. The MRA strategy of \(L_2(0,1)\) consists in representation of the space \(L_2(0,1)\) as a direct sum of some subspaces \(V_n\) and their orthogonal complements \(W_n\). Then, the equation (1) is discretized by applying the projection operators to operators \(A\), \(B\), \(K\), and using new notation some recursion relations are obtained. The steps of the procedure are described and convergence results a presented. After general results, a special type (matrix) Volterra integral equation is treated as an application, using a Haar basis. An algorithm for generating homogenized equations is given. Three numerical examples are discussed and the solutions are compared to other numerical methods.
Mathematics Subject Classification ID: 65J15
Mathematics Subject Classification ID: 47J25
Mathematics Subject Classification ID: 45G15
Mathematics Subject Classification ID: 42C15
Mathematics Subject Classification ID: 65R20
Mathematics Subject Classification ID: 35B27
zbMATH DE Number: 846191
zbMATH Keywords: homogenization
zbMATH Keywords: multiresolution analysis
zbMATH Keywords: Hilbert space
zbMATH Keywords: projection operators
zbMATH Keywords: convergence
zbMATH Keywords: Volterra integral equation
zbMATH Keywords: numerical examples
reviewed by: Iulian Coroian
MaRDI profile type: Publication
full work available at URL: acha.1995.1024
OpenAlex ID: W2026741150
DOI: 10.1006/ACHA.1995.1024
Recommended article: Wavelet-Based Numerical Homogenization
Recommended article: A multiresolution method for numerical reduction and homogenization of nonlinear ODEs
Recommended article: Multiresolution homogenization schemes for differential equations and applications
Recommended article: A comparison of multiresolution and classical one-dimensional homogenization schemes
Recommended article: Q3645657
Recommended article: Exponential decay of the resonance error in numerical homogenization via parabolic and elliptic cell problems
Recommended article: Computation of Quasi-Local Effective Diffusion Tensors and Connections to the Mathematical Theory of Homogenization
Recommended article: On a multiscale computational strategy with time and space homogenization for structural mechanics.
Recommended article: Q5696511
Recommended article: Fast Multiresolution Algorithms for Solving Linear Equations: A Comparative Study
description: scientific article; zbMATH DE number 846191