A convergence proof for a finite element discretization of Chorin's projection method of the incompressible Navier-Stokes equations

TL;DR

This paper proves that a finite element discretization of Chorin's projection method for the incompressible Navier-Stokes equations converges to a weak solution, using energy inequalities and compactness arguments without extra regularity assumptions.

Contribution

It provides a rigorous convergence proof for a finite element scheme of Chorin's method, establishing weak solution convergence under minimal assumptions.

Findings

Convergence of the numerical scheme to a Leray-Hopf weak solution.

Establishment of a discrete energy inequality for the scheme.

Development of a new compactness argument to handle different velocity bounds.

Abstract

We study Chorin's projection method combined with a finite element spatial discretization for the time-dependent incompressible Navier-Stokes equations. The scheme advances the solution in two steps: a prediction step, which computes an intermediate velocity field that is generally not divergence-free, and a projection step, which enforces (approximate) incompressibility by projecting this velocity onto the (approximately) divergence-free subspace. We establish convergence, up to a subsequence, of the numerical approximations generated by this scheme to a Leray-Hopf weak solution of the Navier-Stokes equations, without any additional regularity assumptions beyond square-integrable initial data. A discrete energy inequality yields a priori estimates, which we combine with a new compactness result to prove precompactness of the approximations in $L^2([0,T]\times\Omega)$, where $[0,T]$ is the time interval and $\Omega$ is the spatial domain. Passing to the limit as the discretization parameters vanish, we obtain a weak solution of the Navier-Stokes equations. A central difficulty is that different a priori bounds are available for the intermediate and projected velocity fields; our compactness argument carefully integrates these estimates to complete the convergence proof.