Research On Fully Discrete Finite Element Method For The Voigt Regularization Model

In this paper,we mainly discuss the fully discrete finite element scheme for Kelvin-Voigt model and magnetohydrodynamic-Voigt?MHD-Voigt?model.An important aspect of the Kelvin-Voigt model and the MHD-Voigt model,when used as a regularization for these two equations,is that the regularization is inviscid in the sense that it does not add artificial viscosity.Hence,we refer to the Voigt-regularization as an inviscid reg-ularization.Moreover,the Voigt-regularization can be used to stabilize simulations by a method different from adding artificial viscosity.The main work is organized as fol-lows:In the first part,we study stability and convergence of a fully discrete scheme for the two-dimensional nonstationary Kelvin-Voigt model.This scheme is based on a finite element approximation for space discretization and the Crank-Nicolson-type scheme for time discretization,which is a two step method.We obtain unconditional stability and optimal convergence rate of the considered scheme.At last,the applicability and effec-tiveness of the present algorithm are illustrated by numerical experiments.Furthermore,we verify the differs from the Navier-Stokes model in the sense that it has an additional term??u_t,which takes into account the relaxation property of the fluid.In the second part,we devote the same method as the Kelvin-Voigt model to solve the two-dimensional nonstationary incompressible MHD-Voigt regularization model.We also study stability and convergence of the fully discrete finite element scheme and obtain unconditional sta-bility and optimal convergence rate of velocity and magnetic fields,respectively.Finally,several numerical experiments are investigated to confirm our theoretical findings.We also analysis the convergence of velocity and magnetic fields with varying?₁and?₂.