Justification of the collocation method for the integral equation for a mixed boundary value problem for the Helmholtz equation
type: Item
instance of: scholarly article
title: Justification of the collocation method for the integral equation for a mixed boundary value problem for the Helmholtz equation
zbMATH Open document ID: 1367.65178
DOI: 10.1134/S0965542516070101
author: Elnur H. Khalilov
published in: Computational Mathematics and Mathematical Physics
publication date: 2017-04-04T00:00:00Z
review text: For a bounded domain in \(\mathbb R^3\) with smooth boundary \(S\), the Helmholtz equation is considered along with a third kind boundary condition at \(S\) and the Sommerfeld condition at infinity. Using the fundamental solution and potentials, this problem is rewritten as the determination of a density on \(S\). Proposing appropriate cubature formulas and investigating their accuracy, a discrete problem arises the existence and unicity properties of which are shown. The convergence of the method is analysed invoking a result of \textit{G. M. Vajnikko} [J. Sov. Math. 15, 675--705 (1981; Zbl 0582.65046); translation from Itogi Nauki Tekh., Ser. Mat. Anal. 16, 5--53 (1979)]. The convergence order is shown to depend on a distance characterizing the discretization of \(S\), on a Hölder exponent connected with \(S\), and on the modulus of continuity of the function on the right hand side of the boundary condition. The result generalizes earlier work of \textit{F. A. Abdullaev} and the author [Differ. Equ. 40, No. 1, 89--93 (2004; Zbl 1070.65123); translation from Differ. Uravn. 40, No. 1, 82--86 (2004)]. The investigation is carried out for any wave number but the influence of the latter on the estimates is not shown.
reviewed by: Gisbert Stoyan
Mathematics Subject Classification ID: 65N38
Mathematics Subject Classification ID: 35J05
Mathematics Subject Classification ID: 65N80
Mathematics Subject Classification ID: 65N12
zbMATH DE Number: 6700381
zbMATH Keywords: Helmholtz equation
zbMATH Keywords: mixed boundary value problem
zbMATH Keywords: surface integral equation
zbMATH Keywords: collocation method
zbMATH Keywords: properties of the discretized problem
zbMATH Keywords: convergence estimates
zbMATH Keywords: fundamental solution
zbMATH Keywords: cubature formulas
MaRDI profile type: Publication
cites work: The exterior Dirichlet problem for the Helmholtz equation
cites work: On Neumann's method for the exterior Neumann problem for the Helmholtz equation
cites work: The exterior Neumann problem for the Helmholtz equation
cites work: Justification of the collocation method for a class of boundary integral equations
cites work: Potential-based numerical solution of Dirichlet problems for the Helmholtz equation
cites work: Q5519393
cites work: Q3756626
cites work: Q5628551
cites work: Cubic formula for class of weakly singular surface integrals
cites work: Regular convergence of operators and approximate solution of equations
full work available at URL: s0965542516070101
OpenAlex ID: W2505520812
description: scientific article; zbMATH DE number 6700381