Convergence of the fully discrete incremental projection scheme for incompressible flows

TL;DR

This paper proves the convergence of a fully discrete incremental projection scheme for incompressible Navier-Stokes equations to a weak solution, using finite volume methods on non-uniform meshes without regularity assumptions.

Contribution

It establishes the convergence of both semi-discrete and fully discrete schemes for incompressible flows without requiring regularity of the exact solution.

Findings

Convergence of semi-discrete scheme to a weak solution.

Convergence of fully discrete scheme on non-uniform meshes.

Existence of solutions via a priori estimates.

Abstract

The present paper addresses the convergence of a first order in time incremental projection scheme for the time-dependent incompressible Navier-Stokes equations to a weak solution, without any assumption of existence or regularity assumptions on the exact solution. We prove the convergence of the approximate solutions obtained by the semi-discrete scheme and a fully discrete scheme using a staggered finite volume scheme on non uniform rectangular meshes. Some first a priori estimates on the approximate solutions yield the existence. Compactness arguments, relying on these estimates, together with some estimates on the translates of the discrete time derivatives, are then developed to obtain convergence (up to the extraction of a subsequence), when the time step tends to zero in the semi-discrete scheme and when the space and time steps tend to zero in the fully discrete scheme; the approximate solutions are thus shown to converge to a limit function which is then shown to be a weak solution to the continuous problem by passing to the limit in these schemes.