Shadowing for discrete approximations of abstract parabolic equations
The paper deals with the analysis of a rather general semi-discretization schemes (i.e. with continuous \(t\)) for the nonlinear problem \[ u^{\prime}(t)=Au(t)+f(u(t)), \quad u(0)=u^{0}, \] where \(A\) is an operator coefficient in a Banach space. The authors prove a shadowing theorem which compares the exact and the semi-discrete solutions.The main assumptions can be verified for operators with compact resolvents and for wide classes of the finite-element as well as finite difference discretizations.
scientific article; zbMATH DE number 5302650