Backward Euler discretization of fully nonlinear parabolic problems (Q2759088)

The authors study the backward Euler discretization of the abstract nonlinear evolution equation NEWLINE\[NEWLINEu'(t)= F(u(t)),\quad t> 0,\quad u(0)\in{\mathcal D},NEWLINE\]NEWLINE where \({\mathcal D}\subset D\) denotes an open subset of the Banach space \(D\) that is assumed to be densely embedded in the image Banach space \(X\). The mapping \(F:{\mathcal D}\to X\) is supposed to be Fréchet differentiable on \({\mathcal D}\) with a locally Lipschitz continuous and sectorial Fréchet derivative generating, for every \(U^*\in{\mathcal D}\), a graph norm that is equivalent to the norm in \(D\). The authors provide error estimates for the variable step stepsize backward Euler scheme under different regularity assumptions imposed on the exact solution. Finally, the authors study the behaviour of the backward Euler approximations near an asymptotically stable equilibrium.