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

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−13/2. 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. [ABSTRACT FROM AUTHOR]