Anisotropic Convergence And Error Estimation Problem

This thesis is devoted to the problems of anisotropic finite elements convergence of vicoelastic equation and parabolic integro-differential equation and the error estimates of variable coefficient parabolic equation with anisotropic moving grids finite element methods mainly. It is composed of six chapters.In chapter 1, it is summarized about the historical background and the present progress of anisotropic finite elements. At the same time, the main work of the paper is concluded.In chapter 2, the basic knowledge is introduced, which can be used in the following chapters.In chapter 3, we discuss the convergence of vicoelastic equation with anisotropic nonconforming element. we get the L₂ -error estimate using the anisotropic interpolation theorem, the error estimates of the finite element solution and superclose property of vicoelastic equation. Also we get the superconvergence result on the central point of element of vicoelastic equation using the polynomial test method.In chapter 4, we discuss the convergence of parabolic integro-differential equation with anisotropic triple dimension linear element. Firstly, we prove the integral inequality of the anisotropic grids with a new simlple method—Bilinear Lemma, and we get L₂ -error estimate with Bramble-Hilbert Lemma. The error estimates of the finite element solution, superclose property and global superconvergence property of parabolic integro-differential equation are discussed and derived. At the same time We get the superconvergence result on the central point of element of parabolic integro-differential equation using the polynomial test method.In chapter 5, we discuss the error estimates of variable coefficient parabolic equation with anisotropic moving grids. Firstly, the property of elliptic projection is proved on the anisotropic grids. We get the error estimates formula of the energy norm and L₂ -norm, and the optimal energy norm and L₂ -norm error estimates are derived.In chapter 6, all the work of this thesis is summarized, and the futher work of the anisotropic finite element is given.