A proof of the consistency of the finite difference technique on sparse grids
type: Item
instance of: scholarly article
title: A proof of the consistency of the finite difference technique on sparse grids
zbMATH Open document ID: 0978.65094
author: Yanyan Li
published in: Computing
publication date: 2002-01-21T00:00:00Z
review text: The author shows the equivalence of the finite discretization of differential operators on sparse grids and the extrapolation formula of the combination technique applied to discretized differential operators on rectangular subgrids which correspond to the subspace \(V^I\) for \(I\subset\Lambda\), where \(\Lambda\) is some index set. This subspace consists of the (anisotropic) tensor product of the space \(V^l= \text{span}\{\varphi(2^l x-s)\mid s\in \mathbb Z\}\), where the scaling function \(\phi\) induces a multiresolution analysis. The combination technique extrapolates a very accurate solution on the orthogonal sum of orthogonal complements of \(V^I\), from finite difference or Galerkin solutions on \(V^I\). The above result gives a new proof of the consistency of the sparse grid finite difference technique, thus improving previous results in the following points: weaker smoothness is required, the proof allows for quite general index set \(\Lambda\) and covers higher order discretization and compact finite difference schemes.
reviewed by: Rémi Vaillancourt
Mathematics Subject Classification ID: 65N06
Mathematics Subject Classification ID: 65B05
Mathematics Subject Classification ID: 65N12
Mathematics Subject Classification ID: 65T60
Mathematics Subject Classification ID: 42C40
zbMATH DE Number: 1563392
zbMATH Keywords: finite-difference method
zbMATH Keywords: wavelet
zbMATH Keywords: Galerkin method
zbMATH Keywords: sparse grid
zbMATH Keywords: combination technique
zbMATH Keywords: consistency
zbMATH Keywords: differential operators
zbMATH Keywords: extrapolation
zbMATH Keywords: multiresolution analysis
MaRDI profile type: Publication
full work available at URL: s006070070009
OpenAlex ID: W1990853440
DOI: 10.1007/S006070070009
description: scientific article; zbMATH DE number 1563392