Suitable weak solutions of the Navier–Stokes equations constructed by a space–time numerical discretization

Abstract

We prove that weak solutions obtained as limits of certain numerical space-time discretizations are suitable in the sense of Scheffer and Caffarelli-Kohn-Nirenberg. More precisely, in the space-periodic setting, we consider a full discretization in which the θmethod is used to discretize the time variable, while in the space variables we consider appropriate families of finite elements. The main result is the validity of the so-called local energy inequality.2010 Mathematics Subject Classification. 35Q30,76M10, 76M20.

On the convergence of second-order in time numerical discretizations for the evolution Navier-Stokes equations

“…In this paper, we continue and extend some previous works in [3,6,7,8] and especially [5] to analyze the difficulties arising when dealing with full space-time discretization with schemes which are of second order in the time variable. The aim of this paper is to extend to the case θ = 1/2, which corresponds to the Crank-Nicolson method and which could not be treated directly with the same proofs as in [5]. In particular, the case θ = 1/2 requires a coupling between the space and time mesh-size, which is nevertheless common to other second order models.…”

“…The interplay between suitable weak solutions and numerical computations of turbulent flows has been emphasized starting from the work of Guermond et al [17,18] and a recent overview can be found in the monograph [4]. In this paper, we continue and extend some previous works in [3,6,7,8] and especially [5] to analyze the difficulties arising when dealing with full space-time discretization with schemes which are of second order in the time variable. The aim of this paper is to extend to the case θ = 1/2, which corresponds to the Crank-Nicolson method and which could not be treated directly with the same proofs as in [5].…”

“…We remark that it is a particular case of a more general lemma, whose statement and proof can be found in [26,Lemma 5.1]. Even if it is a tool most often used for compressible equations, it is useful here and we recall the special version taken from [5].…”

“…The rest of the proof follows as in [5], hence we just sketch the proof, referring to that reference for full details.…”

“…The proof of Theorem 1.1 is given in Section 5 and it is based on a compactness argument as we previously developed in [5] and a precise analysis of the quantity u m+1 h − u m h , by using the assumptions linking the time and the spatial mesh size.…”