On the convergence of some numerical method for a hypersingular integral equation

Gleb V. Ryzhakov

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

The Neumann boundary value problem for the Laplace equation arises in the theoretical aerodynamic. We consider the problem in a domain, whose boundary is a smooth closed surface. One of the methods of solve is to represent the solution in the form of a double-layer potential, placed on the surface, and reduce the boundary value problem to the linear hypersingular integral equation. A hypersingular integral treated in the sense of the Hadamard finite value and the integral equation is written out on the surface under consideration. For the integral operator in that equation, we suggest quadrature formulas like the method of vortical frames with a regularization, which provides its approximation on the entire surface for the use of a nonstructured partition. We construct a numerical scheme for the integral equation on the basis of suggested quadrature formulas, prove an estimate for the norm of the pseudoinverse matrix of the related system of linear equations and the uniform convergence of numerical solutions to the exact solution of the hypersingular integral equation on the grid.

Original languageEnglish
Title of host publicationNumerical Analysis and Applied Mathematics, ICNAAM 2012 - International Conference of Numerical Analysis and Applied Mathematics
Pages716-719
Number of pages4
Edition1
DOIs
Publication statusPublished - 2012
Externally publishedYes
EventInternational Conference of Numerical Analysis and Applied Mathematics, ICNAAM 2012 - Kos, Greece
Duration: 19 Sep 201225 Sep 2012

Publication series

NameAIP Conference Proceedings
Number1
Volume1479
ISSN (Print)0094-243X
ISSN (Electronic)1551-7616

Conference

ConferenceInternational Conference of Numerical Analysis and Applied Mathematics, ICNAAM 2012
Country/TerritoryGreece
CityKos
Period19/09/1225/09/12

Keywords

  • boundary value problem
  • fluid flow
  • hypersingular integral
  • integral equation
  • vortex method

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