Convergence of the Piecewise Linear Approximation and Collocation Method for a Hypersingular Integral Equation on a Closed Surfaceстатья
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Аннотация:We study the numerical solution of a linear hypersingular integral equation arising
when solving the Neumann boundary value problem for the Laplace equation by the boundary
integral equation method with the solution represented in the form of a double layer potential.
The integral in this equation is understood in the sense of Hadamard finite value. We construct
quadrature formulas for the integral occurring in this equation based on a triangulation of the
surface and an application of the linear approximation to the unknown function on each of
the triangles approximating the surface. We prove the uniform convergence of the quadrature
formulas at the interpolation nodes as the triangulation size tends to zero. A numerical solution
scheme for this integral equation based on the suggested quadrature formulas and the collocation
method is constructed. Under additional assumptions about the shape of the surface, we prove
a uniform estimate for the error in the numerical solution at the interpolation nodes.