One‐level and multilevel space‐time finite element method for the viscoelastic Kelvin‐Voigt model

Abstract

In this paper, we consider the space‐time finite element method for the viscoelastic Kelvin‐Voigt model. Firstly, based on a priori estimates of spatial semidiscrete numerical solutions, stability and convergence results of one‐level space‐time finite element solutions in different norms are provided under some restrictions on the time step. Secondly, in order to improve the computational efficiency, multilevel space‐time finite element method is introduced. In the multilevel numerical scheme, the nonlinear Kelvin‐Voigt problem is just solved in the coarsest mesh, the Newton iteration is adopted to treat the nonlinear term, and a series of linear Kelvin‐Voigt problems are considered in successive shape conforming regular triangulations. The multilevel space‐time finite element approximate solution is shown to have a convergence rate of the same order as that of the one‐level space‐time finite element solutions of the nonlinear Kelvin‐Voigt problem on a fine mesh with some appropriate choices between the time steps and mesh sizes: kj∼kj−12,hj∼hj−13false/2. Finally, some numerical results are provided to verify the established theoretical findings. Compared with the standard Galerkin finite element method, from the view point of theoretical analysis, the space‐time finite element method has the same convergence orders for variables as the Galerkin method. However, from the view point of numerical simulations, the space‐time finite element method improves the computational accuracy without additional computational cost.

A rotational velocity‐correction projection method for the Kelvin–Voigt viscoelastic fluid equations

“…Considering linearized backward Euler method, Bajpai and Amiya et al derived the optimal error estimates and obtained that the results are valuable when the Kelvin–Voigt converges to the Navier–Stokes system in Kundu et al 12 In Antontsev et al, 13 the authors studied the generalized K‐V system and proved that the weak solution is existent. By using the space‐time finite element method and Euler semi‐implicit scheme, Zhang and Duan 14 obtained the stability and convergence analysis of the Kelvin–Voigt model by using the multilevel space‐time finite element method; at the same time, error estimate of this model was established. In addition, the Galerkin finite element method was also applied to this model, not only the stability of the numerical solution of the spatial discrete scheme is obtained, but also the convergence result is obtained in Zhang and Duan 15 .…”

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

“…Then, they have employed a backward Euler method for the time discretization and have discussed the stability and convergence analysis for the fully discrete approximations. For more developments of numerical methods applied to the model (1.1)- (1.3) and their finite element analysis, one may refer to [2], [29], [37] and the literature mentioned therein. As can be seen, the literature is confined to the finite element analysis for the continuous Galerkin methods applied to the problem (1.1)- (1.3).…”