An unsaturated numerical method for the exterior axisymmetric Neumann problem for Laplace's equation
type: Item
instance of: scholarly article
title: An unsaturated numerical method for the exterior axisymmetric Neumann problem for Laplace's equation
zbMATH Open document ID: 1248.65125
DOI: 10.1134/S0037446611060036
published in: Siberian Mathematical Journal
publication date: 2012-02-24T00:00:00Z
review text: The paper deals with an exterior Neumann boundary value problem for the Laplace equation in \(\mathbb{R}\). Using the simple layer potential the problem is transformed to an equivalent weakly singular integral equation. The last one is discretized by the substituting of the trigonometric interpolation polynomial of the degree \(\leq m\) on an equidistant grid and by the collocation on the same grid. For the resulting system of the linear algebraic equations with respect to the coefficients of the interpolation polynomial an exponentially convergent iteration is proposed. It is shown that the error of the discretization method under consideration (up to some logarithmical factor) is of the order \({\mathcal{O}}(m^{-(k+\alpha)})\) provided that the exact solution belongs to the class \(C^{k+\alpha}(M)\) of \(2 \pi-\)periodic \(k\) times continuously differentiable functions whose \(k-\)th derivative satisfies the Hölder condition with the exponent \(\alpha\) and constant \(M\). In the case of an analytical solution the convergence rate is exponential.
Mathematics Subject Classification ID: 65N38
Mathematics Subject Classification ID: 65N35
Mathematics Subject Classification ID: 65N15
Mathematics Subject Classification ID: 35J05
Mathematics Subject Classification ID: 65N12
zbMATH DE Number: 6009094
zbMATH Keywords: Laplace equation
zbMATH Keywords: simple layer potential
zbMATH Keywords: singular integral equation
zbMATH Keywords: trigonometric interpolation polynomial
zbMATH Keywords: discretization by collocation
zbMATH Keywords: exponential convergence rate
zbMATH Keywords: error bound
zbMATH Keywords: exterior Neumann boundary value problem
author: Vladimir Belykh
reviewed by: Iwan Gawriljuk
MaRDI profile type: Publication
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full work available at URL: s0037446611060036
OpenAlex ID: W2035241054
description: scientific article; zbMATH DE number 6009094