Asymptotic Expansions And Extrapolations For Linear And Quadratic Finite Element Based Common Triangular Meshes

The important part in the finite element high accuracy theory is the asymptotic expansions and extrapolations, which also is the extending and the deepgoing of the high accuracy theory. In this thesis, we analysis the asymptotic expansions and extrapolation of the linear and the quadratic finite elements in some kinds of basic triangular meshs and obtain some important generic characters under the strict smooth condition of function. The method is structural, which use the space orthogonal decomposition in finite element space, the element combination technique, the energy embedding, the separation of variables and the Fourier analysis etc. The main results are as follows.1. In Chapter 2, considering the Criss-Cross linear element with constant coefficient case and variable coefficient case, we obtain the asymptotic expansions and extrapolation of them by using the methods of space orthogonal decomposition and element combination technique. In addition, we get the superconvergence results in the whole region.2. In Chapter 3, considering the different situations of Union Jack and Chevron linear elements, we obtain the characteristic lemma in constructing discrete harmonic complement function by using the methods of energy embedding, separation of variables and Fourier analysis. We also get the regular term parts and the high frequency oscillatory term parts of finite element asymptotic expansions. We respectively get the asymptotic expansions and extrapolation in the two different situations and their high accuracy combination formulas.3. In Chapter 4 and Chapter 5, we discuss the quadratic finite element space with Regular mesh and the one with Criss-Cross mesh. We can respectively obtain the finite element asymptotic expansions and extrapolations in these situations. The high accuracy combination formulas are constructed.