The Finite Element Analysis On Two Class Of Differential Equation Related To Stokes Problem

In this paper, we firstly study the Berandi-Raugel mixed finite element approximation for the integro-differential equation of Stokes type with the semidiscrete scheme. Under the anisotropic meshes, the superclose result is obtained based on higher accuracy analytical technique. At the same time, the globle superconvergence result is also provided through a proper interpolation postprocessing technique. Compared with the general error estimates of the finite element, the convergence rate of velocity u in H~1-norm can be increased from O(h) to O(h~2). Secondly, the Crouzeix-Raviart type anisotropic nonconforming finite element is applied to the same equation with the semidiscrete scheme. In the process, we use interpolation instead of the traditional generalized Ritz-Volterra projection. By novel approaches, the same optimal error estimates are derived as for conforming finite element under regular meshes in previous later rature. Lastly, we construct a new locking-free rectangular nonconforming finite element, which can be used to solve the Stokes problem through penalty method. And by some novel approaches, the optimal error estimates are obtained.