Discrete maximal regularity for the finite element approximation of the Stokes operator and its application

2023/03/28 by Tomoya Kemmochi, Kemmochi, Tomoya
Mathematics · Computer Science · Engineering · #Navier-Stokes equation solutions #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics

paper · pdf · doi:10.48550/arxiv.2303.16236

Abstract

Maximal regularity for the Stokes operator plays a crucial role in the theory of the non-stationary Navier--Stokes equations. In this paper, we consider the finite element semi-discretization of the non-stationary Stokes problem and establish the discrete counterpart of maximal regularity in L^q for q ∈ ( (2N)/(N+2), (2N)/(N-2) ). For the proof of discrete maximal regularity, we introduce the temporally regularized Green's function. With the aid of this notion, we prove discrete maximal regularity without the Gaussian estimate. As an application, we present L^p(0,T;L^q(Ω))-type error estimates for the approximation of the non-stationary Stokes problem.

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