An optimal order collocation method for first kind boundary integral equations on polygons
One of the simplest first kind linear boundary integral equations with a logarithmic kernel, arising in two-dimensional potential theory, is analyzed. It has attracted the intention of many numerical analysts, but despite this a proper explanation of the convergence of the collocation method is still to be given. The authors discuss the convergence of the collocation method using splines of any order \(k\) for such integral equations on closed polygonal boundaries in \(\mathbb{R}^2\). Before discretization the equation is transformed to an equivalent equation over \([-\pi, \pi]\) using a nonlinear parametrization of the polygon. This has the effect of producing a transformed equation with a solution which is smooth on \([- \pi,\pi]\). The structure of the integral operator is described in detail. The authors show that the collocation method using splines of order \(k\) converges with optimal order \(O(h^k)\). Using the numerical solutions of the transformed equation, numerical solutions of the original equation are constructed which converge optimally in a certain weighted norm. The convergence results are illustrated with some numerical examples.
first kind linear boundary integral equation
scientific article; zbMATH DE number 739394