# 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

savare@imati.cnr.it

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

rhn@math.umd.edu

## Abstract

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.