Convergence of the incremental projection method using conforming approximations

TL;DR

This paper proves the convergence of an incremental projection scheme for the Navier-Stokes equations using conforming finite element spaces, ensuring the approximate solutions converge to a weak solution without regularity assumptions.

Contribution

It introduces a convergence proof for an incremental projection method with conforming discretizations, including the construction of a suitable interpolator for Taylor-Hood elements.

Findings

Convergence of the scheme to a weak solution is established.

Existence and uniqueness of the discrete approximation are proven.

The interpolator preserves divergence-free properties in conforming spaces.

Abstract

We prove the convergence of an incremental projection numerical scheme for the time-dependent incompressible Navier--Stokes equations, without any regularity assumption on the weak solution. The velocity and the pressure are discretised in conforming spaces, whose the compatibility is ensured by the existence of an interpolator for regular functions which preserves approximate divergence free properties. Owing to a priori estimates, we get the existence and uniqueness of the discrete approximation. Compactness properties are then proved, relying on a Lions-like lemma for time translate estimates. It is then possible to show the convergence of the approximate solution to a weak solution of the problem. The construction of the interpolator is detailed in the case of the lowest degree Taylor-Hood finite element.