Maximal regularity for time-stepping schemes arising from convolution quadrature of non-local in time equations

Abstract

<p style='text-indent:20px;'>We study discrete time maximal regularity in Lebesgue spaces of sequences for time-stepping schemes arising from Lubich's convolution quadrature method. We show minimal properties on the quadrature weights that determines a wide class of implicit schemes. For an appropriate choice of the weights, we are able to identify the <inline-formula><tex-math id="M1">\begin{document}$ \theta $\end{document}</tex-math></inline-formula>-method as well as the backward differentiation formulas and the <inline-formula><tex-math id="M2">\begin{document}$ L1 $\end{document}</tex-math></inline-formula>-scheme. Fractional versions of these schemes, some of them completely new, are also shown, as well as their representation by means of the Grünwald–Letnikov fractional order derivative. Our results extend and improve some recent results on the subject and provide new insights on the basic nature of the weights that ensure maximal regularity.</p>