Convergence of the piecewise linear approximation and collocation method for a hypersingular integral equation on a closed surface
The paper is concerned with the numerical solution of a linear hypersingular integral equation of the form \[ \int_{S} g(y) \frac{\partial^2 F \left( x-y \right)}{ \partial n_x \partial n_y} d \sigma_y = f\left(x\right), \quad x \in S, \quad F \left( x - y\right) = \frac{1}{\left| x-y \right|}, \] where \(S\) is a smooth closed surface and \(\partial / \partial n_x\) denotes the derivative in the direction of the outward unit normal to \(S\) at the point \(x\). The hypersingular integrals are interpreted in the sense of Hadamard's finite parts. The authors study a collocation method with basis functions that are piecewise linear on the triangulation of the surface \(S\) and they propose quadrature formulas for computing integrals. The uniform convergence of the quadrature formulas at the interpolation nodes is proved and under additional assumptions on the surface \(S\) the uniform error estimates of the proposed method at the interpolation nodes are derived.
scientific article; zbMATH DE number 6819865