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(√τ ).
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 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.