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An efficient and reliable a-posteriori error estimator is developed for a characteristic-Galerkin finite element method for time-dependent convection-dominated problems. An adaptive algorithm with variable time and space steps is proposed and studied. At each time step in this algorithm grid coarsening occurs solely at the final iteration of the adaptive procedure, meaning that only time and space refinement is allowed before the final Communicated by Aihui Zhou. Ming Cui is supported in part by a State Scholarship Fund of China and the Independent 320 M. Cui et al.iteration. It is proved that at each time step this adaptive algorithm is capable of reducing errors below a given tolerance in a finite number of iteration steps. Numerical results are presented to check the theoretical analysis.

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