Anisotropic Finite Element And Mixed Finite Element Analysis

The theme of this paper is the analysis of anisotropic Hermite finite elements and the nonconforming finite elements approximations for some mixed problems.The classical interpolation theory of finite elements is well known.To make the constant in the interpolation error independent of element size,the ratio of the diameter of the element and the diameter of the biggest ball contained in the element,is uniformly bounded.This condition is called regular or nondegenerate condition,which restrict the application of the finite element.In fact,on the one hand,the solution of some practicable problems have anisotropic behvious in the boundary layer or in the corner of the domain, which means that the solution varies significantly only in certain directions.For these problems,the accuracy of the standard finite element methods may decrease.On the other hand,for some problems(for example the complex material problem),the computation amount is very large if we still use the regular mesh.So it is necessary to analyze the anisotropic finite element.Rectantly,many researchers are interested in the anisotropic finite element.In the second chapter,we define the anisotropic C^N-1-continuous Hermite interpolation finite elements on rectangle and cuboid,which are applied in elliptic problems of any 2N order.At last,we carry out the numerical experiments,which support our theoretical analysis.Darcy-Stokes equation models porous media flow coupled with open fluid flow in a single form equation.In some sub-domain,the equation is reduced to the Darcy equation, in some sub-domain,the equation is reduced to the Stokes equation.In a single domain, when v(x)=ε²,α(x)=1,the equation becomes a singular perturbed equation.Because the coeffients in the equation are discontinuous,many standard finite elements are not uniformly stable.In the chapter 3,we construct two new nonconforming rectangle and cuboid elements.We prove the elements are stable for the Darcy-Stokes equation and singular perturbed equation in one domain.For the Stokes equations which goven viscous fluid flow,the natural Galerkin approximation is a standard mixed problem.We should sovle the velocity and pressure at the same time.The key leading to the success of a mixed method is the finite element must satisfy the discrete inf-sup condition or Babuska-Brezzi condition.We define several triangular,rectangular and tetrahedron elements,which are stable for the Stokes problem. The discrete B-B condition holds for the elements,and error estimates in the energy norm for the velocity and L²-norm for the pressure are O(h²).