A study of Nonlinear Galerkin Finite Element for time-dependent incompressible Navier-Stokes equation

TL;DR

This paper investigates nonlinear Galerkin finite element methods for the time-dependent incompressible Navier-Stokes equations, emphasizing the impact of nonlinear terms on convergence and providing improved error estimates in the L2 norm.

Contribution

It introduces and analyzes nonlinear Galerkin methods, demonstrating their convergence properties and deriving optimal error estimates in the L2 norm for linear finite element approximations.

Findings

Nonlinear Galerkin methods significantly influence convergence rates.

Achieved improved error estimates in the L2 norm.

Error bounds are optimal compared to existing H1 norm estimates.

Abstract

In this article, we discuss a couple of nonlinear Galerkin methods (NLGM) in finite element set up for time dependent incompressible Navier-Sotkes equations. We show the crucial role played by the non-linear term in determining the rate of convergence of the methods. We have obtained improved error estimate in $\bL^2$ norm, which is optimal in nature, for linear finite element approximation, in view of the error estimate available in literature, in $\bH^1$ norm.