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An ecient and reliable a posteriori error estimate is derived for linear parabolic equations which does not depend on any regularity assump- tion on the underlying elliptic operator. An adaptive algorithm with variable time-step sizes and space meshes is proposed and studied which, at each time step, delays the mesh coarsening until the nal iteration of the adaptive pro- cedure, allowing only mesh and time-step size renements before. It is proved that at each time step the adaptive algorithm is able to reduce the error indi- cators (and thus the error) below any given tolerance within a nite number of iteration steps. The key ingredient in the analysis is a new coarsening strat- egy. Numerical results are presented to show the competitive behavior of the proposed adaptive algorithm.

An adaptive finite element algorithm with reliable and efficient error control for linear parabolic problems