A posteriori error estimates for fully discrete nonlinear parabolic problems

We present estimates for the spatial error in fully discrete approximations to nonlinear parabolic problems that extend the a posteriori estimates for the continuous time semi-discretization introduced in de Frutos and Novo [J. de Frutos, J. Novo, A posteriori error estimation with the p version of the finite element method for nonlinear parabolic differential equations, Comput. Methods Appl. Mech. Engrg. 191 (2002) 4893–4904]. The error indicator is defined as the difference between the fully discrete Galerkin and the fully discrete postprocessed approximations. We prove that the main term of the temporal error in both the fully discrete Galerkin and the postprocessed approximation is the same. As a consequence, the spatial approximation error can be independently estimated with regard to the time approximation error. Some numerical experiments are provided.