A Priori Error Estimates of a Discontinuous Galerkin Finite Element Method for the Kelvin-Voigt Viscoelastic Fluid Motion Equations

TL;DR

This paper develops and analyzes a discontinuous Galerkin finite element method for Kelvin-Voigt viscoelastic fluid equations, providing optimal a priori error estimates and validating them through numerical experiments, marking a novel contribution in this area.

Contribution

It introduces the first finite element analysis with error estimates for the DG method applied to Kelvin-Voigt fluid equations, including both semi-discrete and fully discrete schemes.

Findings

Optimal error estimates in $L^ abla({f L}^2)$-norm for velocity.

Optimal error estimates in $L^ abla(L^2)$-norm for pressure.

Numerical experiments confirm theoretical error bounds.

Abstract

This paper applies a discontinuous Galerkin finite element method to the Kelvin-Voigt viscoelastic fluid motion equations when the forcing function is in $L^\infty({\bf L}^2)$-space. Optimal a priori error estimates in $L^\infty({\bf L}^2)$-norm for the velocity and in $L^\infty(L^2)$-norm for the pressure approximations for the semi-discrete discontinuous Galerkin method are derived here. The main ingredients for establishing the error estimates are the standard elliptic duality argument and a modified version of the Sobolev-Stokes operator defined on appropriate broken Sobolev spaces. Further, under the smallness assumption on the data, it has been proved that these estimates are valid uniformly in time. Then, a first-order accurate backward Euler method is employed to discretize the semi-discrete discontinuous Galerkin Kelvin-Voigt formulation completely. The fully discrete optimal error estimates for the velocity and pressure are established. Finally, using the numerical experiments, theoretical results are verified. It is worth highlighting here that the error results in this article for the discontinuous Galerkin method applied to the Kelvin-Voigt model using finite element analysis are the first attempt in this direction.