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

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.