An unsaturated numerical method for the exterior axisymmetric Neumann problem for Laplace's equation
The paper deals with an exterior Neumann boundary value problem for the Laplace equation in \(\mathbb{R}\). Using the simple layer potential the problem is transformed to an equivalent weakly singular integral equation. The last one is discretized by the substituting of the trigonometric interpolation polynomial of the degree \(\leq m\) on an equidistant grid and by the collocation on the same grid. For the resulting system of the linear algebraic equations with respect to the coefficients of the interpolation polynomial an exponentially convergent iteration is proposed. It is shown that the error of the discretization method under consideration (up to some logarithmical factor) is of the order \({\mathcal{O}}(m^{-(k+\alpha)})\) provided that the exact solution belongs to the class \(C^{k+\alpha}(M)\) of \(2 \pi-\)periodic \(k\) times continuously differentiable functions whose \(k-\)th derivative satisfies the Hölder condition with the exponent \(\alpha\) and constant \(M\). In the case of an analytical solution the convergence rate is exponential.
scientific article; zbMATH DE number 6009094