Integral-Algebraic Equations: Theory of Collocation Methods II

In a previous paper [SIAM J. Numer. Anal., 51(2013), pp. 2238-2259] we analyzed the optimal orders of convergence of piecewise polynomial collocation solutions for systems of integralalgebraic equations (IAEs) with tractability index μ = 1. The present paper describes the decoupling of systems of IAEs of tractability index μ = 2 and (v+ 1)-smoothing (v ≥ 1). It is then shown that the application of the collocation method to the given system of IAEs is equivalent to the application to the decoupled system. While this in principle forms the basis for an elegant analysis of the optimal order of convergence of the method, we show by an example that collocation is not always feasible for general index-2 IAEs. Following the convergence analysis for semiexplicit index-2 IAEs we present two numerical examples: one to verify the predicted orders of convergence and one to show why the collocation method may break down for general IAEs with μ = 2.