Full asymptotics of spline Petrov-Galerkin methods for some periodic pseudodifferential equations

In this paper we study the existence of a formal series expansion of the error of spline Petrov–Galerkin methods applied to a class of periodic pseudodifferential equations. From this expansion we derive some new superconvergence results as well as alternative proofs of already known weak norm optimal convergence results. As part of the analysis the approximation of integrals of smooth functions multiplied by splines by rectangular rules in analyzed in detail. Finally, some numerical experiments are given to illustrate the applicability of Richardson extrapolation as a means of accelerating the convergence of the methods.