Piecewise Continuous Collocation for Integral Equations

Abstract

The collocation method for integral equations of the second kind is surveyed and analyzed for the case in which the approximate solutions are only piecewise continuous. Difficulties with the usual function space setting of $L_\infty (D)$ are discussed, and a satisfactory sense of point evaluation is given for elements of $L_\infty (D)$. Other approaches which are discussed include (i) reformulation as a degenerate kernel method, (ii) the prolongation-restriction framework of Noble, (iii) other function space settings, and (iv) reformulation as a continuous approximation problem by iterating the piecewise continuous approximate solution in the original integral equation.