A two-level finite element method for time-dependent incompressible Navier-Stokes equations with non-smooth initial data

TL;DR

This paper introduces a two-level finite element method for solving 2D time-dependent incompressible Navier-Stokes equations with non-smooth initial data, achieving optimal convergence despite solution regularity loss at initial time.

Contribution

The paper presents a novel two-level finite element approach that effectively handles non-smooth initial data and provides optimal convergence rates for velocity and pressure.

Findings

Achieves optimal convergence in velocity and pressure norms.

Handles non-smooth initial data with regularity loss at t=0.

Validates the method through rigorous analysis.

Abstract

In this article, we analyze a two-level finite element method for the two dimensional time-dependent incompressible Navier-Stokes equations with non-smooth initial data. It involves solving the non-linear Navier-Stokes problem on a coarse grid of size $H$ and solving a Stokes problem on a fine grid of size $h, h<<H$. This method gives optimal convergence for velocity in $H^1$-norm and for pressure in $L^2$-norm. The analysis takes in to account the loss of regularity of the solution at $t=0$ of the Navier-Stokes equations.