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.