High-precision approximation of mixed finite element method for unsteady Stokes equation

For the two-dimensional unsteady Stokes equations, the Taylor-Hood mixed finite element method is employed for numerical simulation. Firstly, the equation is transformed into the variational form by using the variational principle.Secondly, the domain is uniformly divided into finite triangular elements, and the connection is built among the logical elements and the isoparametric ones. The quadratic basis function is selected for velocity u, and the linear basis function is selected for pressure p, so it establishes a finite element space in spatial scale. Then it is further combined with the finite difference method in temporal scale to construct a fully discrete-type implicit scheme, for selecting the Crank-Nicolson six-point symmetric finite difference scheme is applied. Finally, the equation is transformed into ordinary differential equations for solving its numerical solution, through the numerical example we testify the feasibility and effectiveness of our method. Theoretical constructions and numerical results both validate that the unsteady problem under the space and time discretization it obtains consistent convergent results, which are processed with the linear finite element and the quadratic finite element. And the results of quadratic finite element are much accurate and faster convergence. In order to present the experimental results more intuitively and vividly, this study ends up with several three-dimensional error figures between the exact solution and the finite element solution.