Nonlinear evolution governed by accretive operators in Banach spaces: error control and applications

Giuseppe Savar\'e
Dipartimento di Matematica
Universit\`a di Pavia
and Istituto di Analisi Numerica del C.N.R
27100 Pavia, Italy

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


Nonlinear evolution equations governed by $m$-accretive operators in Banach spaces are discretized via the backward or forward Euler methods with variable stepsize. Computable {\it a posteriori} error estimates are derived in terms of the discrete solution and data, and shown to converge with optimal order $O(\sqrt\tau)$. Applications to scalar conservation laws and degenerate parabolic equations (with or without hysteresis) in $L^1$, as well as to Hamilton-Jacobi equations in $C^0$ are given. The error analysis relies on a comparison principle, for the novel notion of \emph{relaxed solutions}, which combines and simplifies techniques of Benilan and Kru\v zkov. Our results provide a unified framework for existence, uniqueness and error analysis, and yield a new proof of the celebrated Crandall-Liggett error estimate.