On a grid-method solution of the Laplace equation in an infinite rectangular cylinder under periodic boundary conditions
type: Item
instance of: scholarly article
title: On a grid-method solution of the Laplace equation in an infinite rectangular cylinder under periodic boundary conditions
zbMATH Open document ID: 1205.65296
DOI: 10.1134/S0081543810020057
published in: Proceedings of the Steklov Institute of Mathematics
publication date: 2010-11-15T00:00:00Z
review text: The article deals with a new finite difference scheme to approximate the Laplace equation posed on a three dimensional infinite rectangular cylinder with periodic boundary conditions. Under the assumption that the boundary value data is a bounded function and is periodic in the \(x_3\) direction, it is first proved that there exists a unique bounded periodic solution. A special finite difference scheme is suggested. To derive the error estimate, the author proves two results on the exact solution. The first result states that the second derivatives of the exact solution are bounded. The second result states some estimate for the fourth derivatives of the exact solution. Thanks to these two stated results on the exact solution and two other lemmas, the author proves that the convergence order of the finite difference scheme is \(h^2|\,\log\,h|\).
reviewed by: Abdallah Bradji
Mathematics Subject Classification ID: 65N06
Mathematics Subject Classification ID: 65N15
Mathematics Subject Classification ID: 65N12
Mathematics Subject Classification ID: 35J05
zbMATH DE Number: 5816194
zbMATH Keywords: three dimensional infinite rectangular cylinder
zbMATH Keywords: periodic boundary conditions
zbMATH Keywords: Laplace equation
zbMATH Keywords: finite difference scheme
zbMATH Keywords: error bound
zbMATH Keywords: convergence
author: E. A. Volkov
MaRDI profile type: Publication
cites work: Q3280274
full work available at URL: s0081543810020057
OpenAlex ID: W2003955180
description: scientific article; zbMATH DE number 5816194