We present two defect correction schemes to accelerate the Petrov-Galerkin finite element methods  for nonlinear Volterra integro-differential equations. Using asymptotic expansions of the errors, we show that the defect correction schemes can yield higher order approximations to either the exact solution or its derivative. One of these schemes even does not impose any extra regularity requirement, on the exact solution. As by-products, all of these higher order numerical methods can also be used to form a posteriori error estimators for accessing actual errors of the Petrov-Galerkin finite element solutions. Numerical examples are also provided to illustrate the theoretical results obtained in this paper.
- A posteriori error estimators
- Asymptotic expansions
- Defect correction
- Petrov-Galerkin methods
- Volterra integro-differential equations
ASJC Scopus subject areas
- Applied Mathematics