A posteriori error estimates and grid adjustment for a nonlinear parabolic equation

The paper is concerned with a posteriori error estimates needed for the adaptive construction of a space grid in solving an initial-boundary value one-dimensional problem for a nonlinear parabolic partial differential equation by the method of lines. A posteriori error indicators are introduced and studied for the semidiscrete case, and the statement on their convergence rate is presented. Under certain conditions, it adds some more results to those of Moore [SIAM J. Numer. Anal. 31 (1994) 149-164] who treats only a linear equation in the semidiscrete case. Complete statements including proofs will appear as a paper [K. Segeth, A posteriori error estimation with the finite element method of lines for a nonlinear parabolic equation in one space dimension, to appear]. A simple numerical example illustrates the capabilities of a posteriori error estimates when used for grid adjustments in the finite element method of lines. The example fully confirms the theoretical statement. (C) 1999 IMACS/Elsevier Science B.V. All rights reserved.