Anisotropic Finite Element Methods For Parabolic Problem And Variational Inequality With Moving Grid

In this paper, a class of Crouzeix-Raviart type anisotropic nonconforming finite element methods are proposed for parabolic problem and parabolic variational inequality problem with moving grid.We use P₁—nonconforming triangular element to approximate the paroblic problem; and a class of Crouzeix-Raviart type nonconforming elements to approximate the parobolic variational inequality problem, which include five-nodal rectangular element, P₁—rectangular element and P₁—triangular nonconforming element. We proved that all above elements have anisotropic property. By using some novel approaches and techniques, the same optimal error estimates are obtained as the traditional methods. The results of this paper show that is shown that the classical regularity assumption or quasi-uniform assumption on meshes is not necessary to the finite element analysis. Thus the application of finite element can be wided. It is helpful to design adaptive meshes.Moreover, in the previous studies for time-dependent variational inequality problems, with moving grid, the Ritz projection was indispensable in the error analysis. However, with the property of the finite element spaces, we instead the interpolation of Ritz injection directly^[19]. Hence, the proof can be simplified.