Nonconforming Finite Element Methods For The Fourth Order Elliptic Boundary Value And The Fourth Order Singular Perturbation Problems

This thesis mainly discuss the fourth order elliptic boundary value problem and the fourth order elliptic singular perturbation problem. First, for the fourth order elliptic boundary value problem, we propose an abstract convergence theorem, which builds a theoretical framework on how to construct C0-nonconforming elements for the fourth order elliptic problem. It not only offers a new simple practical method for constructing finite element, but also makes that constructing high order convergent finite elements in three dimensional spaces is possible, which fills the research gap in this direction. The basic idea of this method is to divide the shape function space into two subspaces by using bubble functions. One subspace is responsible for the C0-continuity of the shape functions and getting the approximation error. Another one which contains the bubble functions is responsible for the continuity in the mean of the normal derivatives of the shape functions across the elements and getting the consistence error. The resulting element interpolation matrix is a block lower triangular matrix which greatly simplifies the proof of the non-singularity of this matrix.Secondly, by using the bubble functions, we construct some finite elements of second order convergence for different shapes of elements, such as:the triangular element, the rectangular element, the tetrahedral element, the cuboid element and triangular prism element. At the same time, we also present some finite elements of first order convergence for the fourth order elliptic problem in two and three spatial dimensions. On the one hand, it enrich the studies of the nonconforming finite element for the fourth-order elliptic problem. On the other hand, it shows that our method is systemic.Thirdly, we adopt those elements of first order convergence to solve the fourth order elliptic singular perturbation problem and get the uniform convergent result.Finally, we give some numerical examples to check our theoretical results. The thesis consists of six chapters.Chapter1is preface, the historical background of the problems and the significance of this thesis are introduced. The recent developments and some knowledge for finite element are given. And we introduce the Imbedding theorem, Lax-Milgram lemma, Cea lemma, Strang lemma and some inequalities. Chapter2presents an abstract convergence theorem, which builds a theoretical framework on how to construct C0-nonconforming elements for the fourth order ellip-tic problem.In Chapter3, by using bubble function, we construct three elements for plate bending problem. Among them, one element is of first order convergence, the other two are of second order convergence. In addition, we also analyze a triangular element of first order convergence constructed in [59].Chapter4similar to the preceding chapter, we proposed five elements:one tetra-hedral element, two cuboid elements and two triangular prism elements. Two elements of them are of first order convergence, the other three elements are of second order con-vergence. In addition, we also analysis a tetrahedral element of first order convergence constructed in [91].Chapter5describes the research background of the fourth order elliptic singular perturbation problem and adopt those elements of first order convergence to solve the fourth order elliptic singular perturbation problem and get the uniform convergent result.Chapter6is some numerical examples to verify our theoretical results.