Optimal error estimates of the penalty finite element method for the unsteady Navier-Stokes equations with nonsmooth initial data

TL;DR

This paper derives optimal error estimates for penalized finite element methods applied to unsteady Navier-Stokes equations with nonsmooth initial data, combining theoretical analysis and numerical validation.

Contribution

It provides the first rigorous derivation of optimal $L^2$ error estimates for both semidiscrete and fully discrete schemes under realistic conditions.

Findings

Optimal $L^2$ error estimates for velocity and pressure are established.

Numerical examples confirm the theoretical error bounds.

Analysis exploits inverse penalized Stokes operator and negative norm estimates.

Abstract

In this paper, both semidiscrete and fully discrete finite element methods are analyzed for the penalized two-dimensional unsteady Navier-Stokes equations with nonsmooth initial data. First order backward Euler method is applied for the time discretization, whereas conforming finite element method is used for the spatial discretization. Optimal $L^2$ error estimates for the semidiscrete as well as the fully discrete approximations of the velocity and of the pressure are derived for realistically assumed conditions on the data. The main ingredient in the proof is the appropriate exploitation of the inverse of the penalized Stokes operator, negative norm estimates and time weighted estimates. Numerical examples are discussed at the end which conform our theoretical results.