An $$h$$ h – $$p$$ p Version of the Continuous Petrov–Galerkin Finite Element Method for Nonlinear Volterra Integro-Differential Equations

Abstract

We present an $$h$$h---$$p$$p version of the continuous Petrov---Galerkin finite element method for nonlinear Volterra integro-differential equations. We derive a priori error bounds in the $$L^2$$L2- and $$H^1$$H1-norm that are explicit in the time steps, the approximation orders, and the regularity of the exact solution. Numerical experiments are provided to illustrate the theoretical results.