On a three level two-grid finite element method for the 2D-transient Navier-Stokes equations

TL;DR

This paper introduces a three-level two-grid finite element method for 2D transient Navier-Stokes equations, providing optimal error estimates and demonstrating efficiency through numerical experiments.

Contribution

It develops a novel three-step two-grid finite element approach with rigorous error analysis and uniform-in-time estimates for the 2D transient Navier-Stokes equations.

Findings

Optimal error estimates in various norms are established.

The method achieves computational efficiency by linearizing around coarse solutions.

Numerical experiments confirm theoretical results.

Abstract

In this paper, an error analysis of a three steps two level Galekin finite element method for the two dimensional transient Navier-Stokes equations is discussed. First of all, the problem is discretized in spatial direction by employing finite element method on a coarse mesh $\mathcal{T} _H$ with mesh size $H$. Then, in step two, the nonlinear system is linearized around the coarse grid solution, say, $u_H$, which is similar to Newton's type iteration and the resulting linear system is solved on a finer mesh $\mathcal{T}_h$ with mesh size $h$. In step three, a correction is obtained through solving a linear problem on the finer mesh and an updated final solution is derived. Optimal error estimates in $L^{\infty}({\bf L}^2)$-norm, when $h=\mathcal{O} (H^{2-\delta})$ and in $L^{\infty}(\bH^1)$-norm, when $h=\mathcal{O}(H^{4-\delta})$ for the velocity and in $L^{\infty}(L^2)$-norm, when $h=\mathcal{O}(H^{4-\delta})$ for the pressure are established for arbitrarily small $\delta>0$. Further, under uniqueness assumption, these estimates are proved to be valid uniformly in time. Then based on backward Euler method, a completely discrete scheme is analyzed and {\it a priori} error estimates are derived. Finally, the paper is concluded with some numerical experiments.