A posteriori error estimates for the Crank-Nicolson method for parabolic equations

Georgios Akrivis
Computer Science Department
University of Ioannina
451 10 Ioannina, Greece


Charalambos Makridakis
Department of Applied Mathematics
University of Crete
71409 Heraklion-Crete, Greece
and Institute of Applied and Computational Mathematics
FORTH, 71110 Heraklion-Crete, Greece


Ricardo H. Nochetto
Department of Mathematics
University of Maryland, College Park
College Park, MD 20742, USA



We derive a posteriori error estimates for time discretizations by both the Crank--Nicolson and the Crank--Nicolson--Galerkin methods for linear and nonlinear parabolic equations. We provide optimal order estimators of various types. The case of nonsmooth data is also considered. Our approach is based on appropriate Crank--Nicolson reconstructions of the approximate solutions. These functions satisfy two fundamental properties: (i) they are explicitly computable and thus their difference to the numerical solution is controlled a posteriori, and (ii) they lead to optimal order residuals as well as to appropriate pointwise representations of the residuals of the same form as the underlying evolution equation. The resulting estimators are shown to be of optimal order and thus known rates of convergence for the Crank--Nicolson method are recovered.