On a stable calculation of the normal to a surface given approximately

Abstract

The paper proposes a stable method for constructing a normal to a surface given approximately. The normal is calculated as the gradient of the function in the surface equation. As is known, the problem of calculating the derivative is ill-posed. In the paper, an approach is adopted to solving this problem as to the problem of calculating the values of an unbounded operator. To construct its stable solution, the principle of minimum of the smoothing functional in Morozov’s formulation is used. The normal is obtained in the form of a Fourier series in the expansion in terms of eigenfunctions of the Laplace operator in a rectangle with boundary conditions of the second kind. The functional stabilizer uses the Laplacian, which makes it possible to obtain a normal in the form of a Fourier series that converges uniformly to the exact normal vector as the error in the surface definition tends to zero. The resulting approximate normal vector can be used to solve various problems of mathematical physics using surface integrals, normal derivatives, simple and double layer potentials.