Approximation of viscosity solutions of elliptic partial differential equations on minimal grids
type: Item
instance of: scholarly article
title: Approximation of viscosity solutions of elliptic partial differential equations on minimal grids
zbMATH Open document ID: 0846.65053
author: Maciej Kocan
published in: Numerische Mathematik
publication date: 1996-09-26T00:00:00Z
review text: Estimates in terms of the ellipticity are obtained for the minimal size \(N_A\) of an equally spaced grid used in the approximation of solutions to elliptic partial differential equations. Because of the equal spacing, the results involve the theory of Diophantine approximation and particularly continued fractions. A general upper bound for the minimal size is established for the strictly elliptic problem by approximating orthogonal bases in \(\mathbb{R}^n\) by orthogonal bases with integer coordinates [this result is proved separately in an Appendix by \textit{W. M. Schmidt}, ibid. 72, No. 1, 117-122 (1995; reviewed below)]. In the two-dimensional case, the minimal size is infinite or \(\max\{n, m\}\) according as the ratio of the diagonal terms of the matrix \(A\) is irrational or \(n/m\) (\(m\), \(n\) coprime). Bounds for \(N_A\) are obtained in the two-dimensional case when the matrix \(A\) of the differential equation is of the form \[ A= (\begin{smallmatrix} 1\\ \sqrt\alpha\end{smallmatrix} \begin{smallmatrix} \sqrt\alpha\\ \alpha+ \varepsilon\end{smallmatrix}), \] \(\alpha, \varepsilon> 0\), \(\varepsilon\) small and \(\alpha\neq 1\). When \(\sqrt\alpha\) is irrational, \(N_A\leq C/\varepsilon\) and when \(\sqrt\alpha\) is badly approximable, \(N_A\leq C/\sqrt\varepsilon\). The last estimate cannot be improved in general although when \(\sqrt\alpha\) is badly approximable, \(N_A\geq C\varepsilon^{- 1/4}\), an estimate which also holds for irrational \(\sqrt\alpha\) for an infinite sequence of \(\varepsilon\).
Mathematics Subject Classification ID: 65N06
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 11J70
zbMATH DE Number: 839149
zbMATH Keywords: viscosity solutions
zbMATH Keywords: elliptic partial differential equations
zbMATH Keywords: minimal grids
zbMATH Keywords: Diophantine approximation
zbMATH Keywords: continued fractions
reviewed by: M. M. Dodson
MaRDI profile type: Publication
full work available at URL: s002110050160
OpenAlex ID: W2010634241
DOI: 10.1007/S002110050160
description: scientific article; zbMATH DE number 839149