A posteriori error estimates for variable time-step discretizations of nonlinear evolution equations

We study the backward Euler method with variable time steps for abstract evolution equations in Hilbert spaces. Exploiting convexity of the underlying potential or the angle-bounded condition, thereby assuming no further regularity, we derive novel a posteriori estimates of the discretization error in terms of computable quantities related to the amount of energy dissipation or monotonicity residual. These estimators depend solely on the discrete solution and data and impose no constraints between consecutive time steps. We also prove that they converge to zero with an optimal rate with respect to the regularity of the solution. We apply the abstract results to a number of concrete, strongly nonlinear problems of parabolic type with degenerate or singular character. (C) 2000 John Wiley & Sons, Inc.