Convergence of the Method of Piecewise Linear Approximations and Collocations for a Two-Dimensional Hypersingular Integral Equation on a Set with Boundary

We consider a hypersingular integral equation on a convex bounded set on the plane withan integral understood in the sense of Hadamard finite part. In particular, equations of this typearise when solving the Neumann boundary value problem for the Laplace and Helmholtz equationson a flat screen in the case where the solution is sought in the form of a double layer potential. Tonumerically solve the equation, we use a numerical scheme based on piecewise linearapproximation of the unknown function on a triangular conformal grid and the collocationmethod. The uniform convergence of numerical solutions to an exact solution on a grid when themaximum cell diameter tends to zero has been proven.