Adv Comput Math (2012) 37:319–353 DOI 10.1007/s10444-011-9210-7 Reliable and efficient error control for an adaptive Galerkin-characteristic method for convection- dominated diffusion problems Ming Cui · Zhangxin Chen · Richard E. Ewing · Guan Qin · Hongsen Chen Received: 2 February 2008 / Accepted: 26 January 2010 / Published online: 17 September 2011 © Springer Science+Business Media, LLC 2011 Abstract An efficient and reliable a-posteriori error estimator is devel- oped 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 Innovation Foundation of Shandong University, IIFSDU. M. Cui School of Mathematics and System Science, Shandong University, Jinan, 250100, China e-mail: mingcui@sdu.edu.cn Z. Chen (B ) · H. Chen Department of Chemical & Petroleum Engineering, Schulich School of Engineering, University of Calgary, 2500 University Drive N.W., Calgary, AB T2N 1N4, Canada e-mail: zhachen@ucalgary.ca H. Chen e-mail: hchenyz@yahoo.com R. E. Ewing Institute for Scientific Computation, Texas A & M University, College Station, TX 77843-3404, USA G. Qin Chemical and Petroleum Engineering Department, University of Wyoming, 1000 E. University Ave., Dept. 3295, Laramie, WY 82071-3295, USA e-mail: guan.l.qin@gmail.com