Discrete maximal regularity for abstract Cauchy problems
Discrete maximal regularity for abstract Cauchy problems
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DOI：
10.4064/sm8495-7-2016
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发表时间：
2016
影响因子：
0.8
通讯作者：
T. Kemmochi
中科院分区：
文献类型：
--
作者：
T. Kemmochi
极大正则性是非线性偏微分方程组理论中的一个基本概念，例如拟线性抛物型方程和NavierStokes方程。因此，当方程被离散化以进行数值计算时，自然会问这种概念的离散类比是否成立。本文引入了有限差分法(θ方法)的离散极大正则性的概念，并证明了离散极大正则性与有界算子的(连续)极大正则性大致等价。此外，我们还证明了在向后欧拉方法的情况下，这种刻画对于无界算子也是成立的。
Maximal regularity is a fundamental concept in the theory of nonlinear partial differential equations, for example, quasilinear parabolic equations, and the NavierStokes 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 (θ-method), and show that discrete maximal regularity is roughly equivalent to (continuous) maximal regularity for bounded operators. In addition, we show that this characterization is also true for unbounded operators in the case of the backward Euler method.