The Finite Element Methods For Fourth Order Elliptic Equations In Three Spatial Dimensions And Incompressible Fluids

In this paper, two new nonconforming tetrahedral elements are presented for the fourth order elliptic partial differential operators in three spatial dimensions. Of the two newly constructed elements one is the 14-parameter tetrahedral element which has lower degree, the other is the 16-parameter energy-orthogonal tetrahedral element which has an energy-orthogonal shape function space. These two elements are proved to be convergent for a model biharmonic equation in three dimensions.In the second part of this paper, we constructed a 20-parameter triangular prism element for the fourth order elliptic equation and analysed the approximation of this element.For the stationnary Stokes problem, when the pressure space is taken as Q = {q∈H1 (Ω);∫Ωqdx= 0}, we get the new variational form. The mixed element formats of any order based on bubble func-tions are presented for the new variational problem in triangular and tetrahedral meshes. and the convergence of any order mixed element formats are proved. Furthermore, when the pressure finite element space consists piecewise polynomials of degree two, we propose the new mixed element formats of one order, two order and three order based on bubble functions for the new variational problem, the error estimates of the pressure in L2-norm for the new formats are one degree higher than that of the existing formats.At the end of this paper, by the macroelement partition theorem, we study the uniform stability and approximation properties of P2-P1 element for Darcy-Stokes problem on irregular crisscross meshes, on distorted crisscross meshes and on barycentric trisected meshes respectively.