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

TL;DR

This paper proves the convergence of second-order time discretization methods, specifically Crank-Nicolson combined with finite elements, to physically relevant weak solutions of the 3D Navier-Stokes equations, ensuring local energy inequalities.

Contribution

It demonstrates convergence of certain second-order numerical schemes to weak solutions of Navier-Stokes equations satisfying local energy inequalities, without requiring extra regularity assumptions.

Findings

Convergence of Crank-Nicolson finite element schemes to weak solutions.

Validation of schemes satisfying local energy inequalities.

Identification of schemes providing alternative existence proofs.

Abstract

We prove the convergence of certain second-order numerical methods to weak solutions of the Navier-Stokes equations satisfying in addition the local energy inequality, and therefore suitable in the sense of Scheffer and Caffarelli-Kohn-Nirenberg. More precisely, we treat the space-periodic case in three space-dimensions and we consider a full discretization in which the the classical Crank-Nicolson method ($\theta$-method with $\theta=1/2$) is used to discretize the time variable, while in the space variables we consider finite elements. The convective term is discretized in several implicit, semi-implicit, and explicit ways. In particular, we focus on proving (possibly conditional) convergence of the discrete solutions towards weak solutions (satisfying a precise local energy balance), without extra regularity assumptions on the limit problem. We do not prove orders of convergence, but our analysis identifies some numerical schemes providing also alternate proofs of existence of "physically relevant" solutions in three space dimensions.