Variable time-step ϑ-scheme for nonlinear evolution equations governed by a monotone operator.

The single-step ϑ-scheme on a variable time grid is employed for the approximate solution of the initial-value problem for a nonlinear first-order evolution equation. The evolution equation is supposed to be governed by a possibly time-dependent hemicontinuous operator that is (up to a shift) monotone and coercive, and fulfills a certain growth condition. A piecewise constant as well as piecewise linear prolongation of the time-discrete solution is shown to converge towards the exact solution if ϑ≥1/2 (including the Crank-Nicolson scheme). In the appearance of a strongly continuous perturbation of the monotone main part, the method is still convergent if ϑ>1/2 and if the ratio of adjacent step sizes is bounded from above by a power of ϑ/(1− ϑ). Besides convergence also well-posedness of the time-discrete problem as well as a priori error estimates are studied. [ABSTRACT FROM AUTHOR]