Nonlinear Galerkin Finite Element for Viscoelastic Fluid Flow: Optimal Error Estimate

TL;DR

This paper develops an improved nonlinear Galerkin finite element method for 2D viscoelastic fluid flow, providing optimal error estimates that enhance the accuracy of linear finite element approximations in Oldroyd models.

Contribution

It introduces a nonlinear Galerkin approach with optimal error estimates for viscoelastic fluid flow, advancing the theoretical understanding of finite element methods in this context.

Findings

Achieved optimal error estimates in $L^{ abla}(L^2)$ norm.

Enhanced the accuracy of finite element approximations for Oldroyd models.

Provided theoretical validation for the nonlinear Galerkin method.

Abstract

In this article, we discuss a couple of nonlinear Galerkin method (NLG) in finite element set up for viscoelastic fluid flow, mainly equations of motion arising in the flow of 2D Oldroyd model. We obtain improved error estimate in $L^{\infty}(\bL^2)$ norm, which is optimal in nature, for linear finite element approximation, in view of the error estimate available in literature, in $L^2(\bH^1)$ norm.