On the convergence of a numerical method for a hypersingular integral equation on a closed surface
We consider a linear integral equation with a hypersingular integral treated in the sense of the Hadamard finite value. This equation arises in the solution of the Neumann boundary value problem for the Laplace equation with a representation of a solution in the form of a double-layer potential. We consider the case in which the interior or exterior boundary value problem is solved in a domain whose boundary is smooth closed surface, and an integral equation is written out of that surface. For the integral operator in that equation, we suggest quadrature formulas like the method of vortical frames with a regularisation, 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 inverse matrix of related system of linear equations and the uniform convergence of numerical solutions to the exact solution of the hypersingular integral equation on the grid.
scientific article; zbMATH DE number 5873488