Discrete maximal regularity for abstract Cauchy problems
Maximal regularity is a fundamental concept in the theory of nonlinear partial differential equations, for example, quasilinear parabolic equations, and the Navier–Stokes equations. It is thus natural to ask whether the discrete analogue of this notion holds when the equation is discretized for numerical computation. In this paper, we introduce the notion of discrete maximal regularity for the finite difference method ($\theta $-method), and show that discrete maximal regularity is roughly equivalent to (continuous) maximal regularity for bounded operators in the case of UMD spaces. The feature of our result is that it includes the conditionally stable case ($0 \le \theta \lt 1/2$). We pay close attention to the dependence of the constants appearing in estimates. In addition, we show that this characterization is also true for unbounded operators in the case of the backward Euler method.