Collocation methods for second kind integral equations with non-compact operators
The authors analyze the uniform convergence of collocation solutions in the space of piecewise continuous polynomial splines of degree r-1 for linear integral equations of the second kind on the half line, where the integral operator is a compact perturbation of a Wiener-Hopf operator. If the underlying mesh is suitably graded, i.e. if the mesh points \(\{s_ i:\) \(i=1,...,n\}\) are given by \(s_ i=(r/\mu)\cdot \ell n(m/(m+1-i))\) (for appropriately chosen \(m\geq n\), and with the real number \(\mu >0\) characterizing the exponential decay of the exact solution at infinity), then optimal order of convergence holds for the collocation solution. Optimal superconvergence results for the corresponding iterated collocation solution are established on analogously graded meshes. These convergence results are then extended to boundary integral equations on polygonal domains. The paper generalizes and complements a number of results contained by the second author and \textit{A. Spence} [IMA J. Numer. Anal. 6, 153-172 (1986; Zbl 0593.65095)] and by \textit{G. A. Chandler} and \textit{I. G. Graham} [Math. Comp. 50, 125-138 (1988; Zbl 0639.65074)].
linear integral equations of the second kind on the half line
scientific article; zbMATH DE number 4153847