Petrov-Galerkin methods for nonlinear volterra integro-differential equations

This paper presents a class of Petrov-Galerkin finite element (PGFE) method s for the initial-value problem for nonlinear Volterra integro-differential equations: y ' (t) = f(t, y(t) + integral (t)(0) k(t,s,y(s))ds, t is an element of I:= [0,T], y(0)=0. These methods have global optimal convergence rates, and have certain globa l and local super-convergence features. Several post-processing techniques are proposed to obtain globally super-convergent approximations. As by prod ucts, these super-convergent approximations can be used as efficient a-post eriori error estimators. Numerical examples are provided to illustrate prop erties of these methods.