On the convergence of the vortex loop method with regularization for the Neumann boundary value problem on a plane screen
The Laplace equation in \(\mathbb{R}^3\) along with Neumann conditions on a finite plane surface \(\Sigma\) is considered for Hölder-continuous Neumann data (with Hölder exponent \(\alpha\)). This exterior boundary value problem is reduced to a singular integral equation on \(\Sigma\) the singularity of which is understood in the Hadamard sense. A collocation method for the numerical solution of this integral equation is proposed which uses a partition of \(\Sigma\) into finite elements, and piecewise constant basis functions. (The physical interpretation of this approach leads to the name of the method mentioned in the title.) The authors prove the uniform convergence of their approach and obtain an error estimate in case \(\alpha>\frac12\).
scientific article; zbMATH DE number 5998581