Variable time-step \(\vartheta\)-scheme for nonlinear evolution equations governed by a monotone operator (Q734131)

This paper is concerned with time-discretisation of the initial-value problems for nonlinear evolution equation of the form \(u'+Au=f\) in \((0, T)\), \(u(0)=u_0\) and its generalizations. Here \(A\) is a Nemytskii operator corresponding to a family of hemicontinuous operators. In this paper it is proved the convergence of piecewise polynomial prolongations of the discrete numerical solutions towards the weak solution without requiring any additional regularity of the exact solution. The method of proof relies on the theory of monotone operators and ompactness arguments combined with algebraic relations that describe properties of the temporal discretization.