Fourth-order Elliptic Equations, Nonconforming Finite Element Method

In this thesis,several nonconforming finite element methods for plate problems and fourth order elliptic singular perturbation problems are discussed.For the reason of technique difficulties,nonconforming plate elements are always employed to solve plate bending problems.Many successful plate elements have been constructed,however, they are not always convergent for fourth order singular perturbation problems.On the other hand,most of the discussions are based on regular condition or quasi-uniform assumption of triangulations.Here,we focus on the convergence analysis of different kinds of nonconforming finite element approximations to fourth order elliptic problems with different mesh fashions.In Chapter 1,we introduce some definitions and notations of basic spaces,present the fundamental knowledge and relational properties of finite element method,such as some important notions,inequalities and useful theorems.In Chapter 2,the approximation of plate bending problem by employing two Morley-type non-C~0 nonconforming plate elements under anisotropic meshes are discussed, the optimal anisotropic interpolation error and consistency error estimates are obtained by using some novel approaches.Some numerical tests are given to confirm the theoretical analysis.In Chapter 3,a counterexample is given to show that the first element in Chapter 2 diverges for the reduced second order equations,which means that it is not convergent for fourth order elliptic singular perturbation problems uniformly.As an alternative,the convergence in the energy norm uniformly with respect to the perturbation parameter is derived when a modified finite element approximation is employed.Numerical experiments are carried out to confirm our theoretical results.Moreover,by employing some new tricks,we obtain the convergence results of the second element in Chapter 2 even when the regular condition or quasi-uniform assumption on the triangulation is not satisfied.Since this element is also non-C~0,this means that the convergence does not require the finite element space being a subspace of H~1(Ω).Similar estimates are also presented for other two different approximation formulations.Numerical results are also given at the end of this chapter.In Chapter 4,a general convergence theorem for C~0 nonconforming finite element to solve the elliptic fourth order singular perturbation problem is presented.The error estimates only related to the righthand term of the equation are also derived when there exist boundary layers.Then the properties of a nine parameter triangular element constructed by double set method is studied,the corresponding convergence results are obtained by the former theorem,numerical experiments are carried out to check our analysis results.