Estimation of the convergence rate of difference solutions to generalized solutions of the Dirichlet problem for the Helmholtz equation of class \(\overset\circ W^ 1_ 2(\Omega)\) in a convex domain
type: Item
instance of: scholarly article
title: Estimation of the convergence rate of difference solutions to generalized solutions of the Dirichlet problem for the Helmholtz equation of class \(\overset\circ W^ 1_ 2(\Omega)\) in a convex domain
zbMATH Open document ID: 0607.65059
DOI: 10.1007/BF01056766
author: S. A. Vojtsekhovskij
author: T. G. Shablij
author: Vladimir L. Makarov
published in: Ukrainian Mathematical Journal
publication date: 1986-01-01T00:00:00Z
review text: For the Helmholtz equation with Dirichlet condition in a convex polygon \(\Omega\), a difference scheme was constructed by the authors and \textit{A. A. Samarskij} [Dokl. Akad. Nauk SSSR 273, 1040-1044 (1983; Zbl 0556.65080)] which, on any nonuniform mesh \({\hat \omega}\) in mesh norm \(L_ 2({\hat \omega})\), has accuracy O(h) if the solution u(x) of the initial problem belongs to space \(\overset\circ W^ 1_ 2(\Omega)\). In the present article we take the case when \(\Omega\) is any convex domain with boundary \(\Gamma \in C^ 2\). By meas of operators of exact difference schemes, we construct a scheme whose solution converges to the solution of the initial problem at a rate \(O(h^{1/2})\) in mesh norm \(L_ 2(\Omega)\).
Mathematics Subject Classification ID: 65N12
Mathematics Subject Classification ID: 65N06
Mathematics Subject Classification ID: 35J05
zbMATH DE Number: 3982443
zbMATH Keywords: convergence rate
zbMATH Keywords: generalized solutions
zbMATH Keywords: Dirichlet problem
zbMATH Keywords: Helmholtz equation
MaRDI profile type: Publication
cites work: Q3960076
cites work: Projection Methods for Dirichlet's Problem in Approximating Polygonal Domains with Boundary-Value Corrections
cites work: Bounds for a class of linear functionals with applications to Hermite interpolation
description: scientific article; zbMATH DE number 3982443