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

Comm. Pure Appl. Math., 53 (2000), 525-589.

Giuseppe Savare
Istituto di Analisi Numerica del C.N.R.
Via Abbiategrasso 209
27100 Pavia, Italy


Ricardo H. Nochetto
Department of Mathematics
University of Maryland, College Park
College Park, MD 20742, USA


Claudio Verdi
Dipartimento di Matematica
Universita di Milano
Via Saldini 50
20133 Milano, Italy



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 \textit{a posteriori} estimates of the discretization error in terms of computable quantities related to the amount of energy dissipation or monotonicity residual. These estimators solely depend 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.