Nonlinear Evolution Governed by Accretive Operators in Banach Spaces: Error Control and Applications

Nonlinear evolution equations governed by $m$-accretive operators in Banach spaces are discretized via the backward or forward Euler methods with variable stepsize. Computable a posteriori error estimates are derived in terms of the discrete solution and data, and shown to converge with optimal order $O(sqrttau)$. 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 Kruzkov. Our results provide a unified framework for existence, uniqueness and error analysis, and yield a new proof of the celebrated Crandall-Liggett error estimate