Digital Object Identifier (DOI) 10.1007/s002119900065 Numer. Math. (1999) 83: 455–475 Numerische Mathematik c  Springer-Verlag 1999 A posteriori error estimation with the finite element method of lines for a nonlinear parabolic equation in one space dimension ⋆ K. Segeth Mathematical Institute of the Academy of Sciences, ˇ Zitn´ a 25, CZ-115 67 Praha 1, Czech Republic; e-mail: segeth@math.cas.cz, Fax +4202-22211638 Received June 15, 1997 / Revised version received May 15, 1998 / Published online: June 29, 1999 Summary. Convergence of a posteriori error estimates to the true error for the semidiscrete finite element method of lines is shown for a nonlinear parabolic initial-boundary value problem. Mathematics Subject Classification (1991): 65M15, 65M20 1. Introduction Adaptive methods for solving parabolic equations are mostly based on a posteriori error estimates [1, 2, 3, 5, 7, 8, 9, 10, 12]. One of the most common strategies for constructing such estimates is the finite element p-refinement, i.e. the computation of a second, higher order solution. The error estimate can be computed as a correction to the original solution on each element. This approach is very suitable for solving parabolic partial differential equations by the method of lines. The analysis of the approximate solution at the actual time level based on the calculation of an a posteriori error estimate yields a new grid to be used for the time step leading to the next time level. Experimental evidence indicates a high efficiency of this approach to linear as well as nonlinear problems. Convergence of the error estimates to the true error for a semidiscrete method has been shown for linear equations in [3, 8, 12]. The paper [9] is devoted to semidiscrete error estimation in the semilinear case and fully discrete error estimation (if SIRK or BDF methods are used) in the nonlinear case. In the present paper, we partially extend the semidiscrete error estimation to the nonlinear case. The results ⋆ This research was supported by the Grant Agency of the Czech Republic under Grant No. 201/97/0217