Discrete maximal regularity for abstract Cauchy problems

Tomoya Kemmochi¹

¹ Graduate School of Mathematical Sciences The University of Tokyo Komaba 3-8-1, Meguro-ku Tokyo, 153-8914, Japan URL: <a href="http://t-kemmochi.github.io/">http://t-kemmochi.github.io/</a>

Studia Mathematica, Tome 234 (2016) no. 3, pp. 241-263

Résumé

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.

Détail

DOI : 10.4064/sm8495-7-2016

Keywords: maximal regularity fundamental concept theory nonlinear partial differential equations example quasilinear parabolic equations navier stokes equations natural ask whether discrete analogue notion holds equation discretized numerical computation paper introduce notion discrete maximal regularity finite difference method theta method discrete maximal regularity roughly equivalent continuous maximal regularity bounded operators umd spaces feature result includes conditionally stable theta pay close attention dependence constants appearing estimates addition characterization unbounded operators the backward euler method