Superapproximation Of Finite Element Methods For Three-dimensional Problems

In this thesis, we mainly study superapproximation theory of finite element methods for three-dimensional problems. For convenience, we only consider Dirichlet boundary value problems of Poisson equations. By means of estimates for three-dimensional discrete derivative Green's function, the theory of the interpolation operator of projection type and the technique of cancellation of elements, for several usual elements in the three-dimensional setting, including rectangular parallelepiped elements, tetrahedral elements and triangular prism elements, we discuss maximum-norm superapproximations of the gradients of finite element solutions in detail, and obtain superapproximation results with high accuracy. This thesis is arranged as follows:In Chapter 1, we introduce some elementary theorems, usual notations and model problems needed in other chapters.In Chapter 2, we introduce the theory of multidimensional discrete Green's function, which plays a crucial role in the study of the superconvergence (especially, the maximum-norm superconvergence) for multidimensional finite elements. Up to now, due to the lack of this theory and the complexity of multidimensional problems themselves, superconvergent results are relatively scarce in the multidimensional case. With the establishment of this theory, it will be easy for one to study the superconvergence for multidimensional problems.In Chapter 3, we introduce the theory of the three-dimensional interpolation operator of projection type, which is an important means to study the superconvergence of the rectangular parallelepiped finite element.In Chapter 4, we discuss superapproximation of the rectangular parallelepiped finite element. For tensor-product rectangular parallelepiped elements and several usual serendipity elements, we derive weak estimates and obtain maximum-norm superapproximate results of the gradients.In Chapter 5, we discuss superapproximation of the tetrahedral finite element. Using the technique of cancellation of elements, we derive the weak estimate of the first type for quadratic tetrahedral elements. Furthermore, maximum-...