Optimal first-order error estimates of a fully segregation scheme for the Navier-Stokes equations

TL;DR

This paper presents an optimal first-order error analysis of a fully discrete, decoupled scheme for 3D Navier-Stokes equations using finite elements, demonstrating optimal convergence without mesh or time step restrictions.

Contribution

It introduces a fully discrete, decoupled scheme with proven optimal error estimates for 3D Navier-Stokes equations, improving understanding of error behavior in segregated methods.

Findings

Optimal error estimates of order O(k+h) are established.

The scheme achieves convergence without constraints on mesh size and time step.

Numerical results confirm theoretical predictions and compare favorably with other schemes.

Abstract

A first-order linear fully discrete scheme is studied for the incompressible time-dependent Navier-Stokes equations in three-dimensional domains. This scheme, based on an incremental pressure projection method, decouples each component of the velocity and the pressure, solving in each time step, a linear convection-diffusion problem for each component of the velocity and a Poisson-Neumann problem for the pressure. Using first-order \emph{inf-sup} stable $C^0$-finite elements, optimal error estimates of order $O(k+h)$ are deduced without imposing constraints on $h$ and $k$, the mesh size and the time step, respectively. Finally, some numerical results are presented according the theoretical analysis, and also comparing to other current first-order segregated schemes.