TY - JOUR

T1 - On the convergence of the vortex loop method with regularization for the Neumann boundary value problem on a plane screen

AU - Ryzhakov, G. V.

AU - Setukha, A. V.

PY - 2011/9

Y1 - 2011/9

N2 - We consider a linear integral equation with a supersingular integral treated in the sense of the Hadamard finite value, which arises in the solution of the Neumann boundary value problem for the Laplace equation with the representation of the solution in the form of a doublelayer potential. We consider the case in which the exterior boundary value problem is solved outside a plane surface (a screen). For the integral operator in the above-mentioned equation, we suggest quadrature formulas of the vortex loop method with regularization, which provide its approximation on the entire surface when using an unstructured partition. In the problem in question, the derivative of the unknown density of the double-layer potential, as well as the errors of quadrature formulas, has singularities in a neighborhood of the screen edge. We construct a numerical scheme for the integral equation on the basis of the suggested quadrature formulas and prove an estimate for the norm of the inverse matrix of the resulting system of linear equations and the uniform convergence of the numerical solutions to the exact solution of the supersingular integral equation on the grid.

AB - We consider a linear integral equation with a supersingular integral treated in the sense of the Hadamard finite value, which arises in the solution of the Neumann boundary value problem for the Laplace equation with the representation of the solution in the form of a doublelayer potential. We consider the case in which the exterior boundary value problem is solved outside a plane surface (a screen). For the integral operator in the above-mentioned equation, we suggest quadrature formulas of the vortex loop method with regularization, which provide its approximation on the entire surface when using an unstructured partition. In the problem in question, the derivative of the unknown density of the double-layer potential, as well as the errors of quadrature formulas, has singularities in a neighborhood of the screen edge. We construct a numerical scheme for the integral equation on the basis of the suggested quadrature formulas and prove an estimate for the norm of the inverse matrix of the resulting system of linear equations and the uniform convergence of the numerical solutions to the exact solution of the supersingular integral equation on the grid.

UR - http://www.scopus.com/inward/record.url?scp=81355142835&partnerID=8YFLogxK

U2 - 10.1134/S0012266111090138

DO - 10.1134/S0012266111090138

M3 - Article

AN - SCOPUS:81355142835

VL - 47

SP - 1365

EP - 1371

JO - Differential Equations

JF - Differential Equations

SN - 0012-2661

IS - 9

ER -