Superconvergence of a Collocation-type Method for Hummerstein Equations

This paper considers the numerical solution of Hammerstein equations of the form $y(t)=f(t)+∫abk(t,s)g(s,y(s))ds,t∈[a,b],$ by a collocation method applied not to this equation, but rather to an equivalent equation for z(t) :=g(t, y(t)). The desired approximation to y is then obtained by use of the (exact) equation $y(t)=f(t)+∫abk(t,s)z(s)ds,t∈[a,b].$ In an earlier paper, questions of existence and optimal convergence of the respective approximations to z and y were examined. In this sequel, collocation approximations to z are sought in certain piecewise polynomial function spaces, and analogous of known superconvergence results for the iterated collocation solution of (linear) second-kind Fredhoim integral equations are stated and proved for the approximation to y.