The Optimal Superconvergence Postprocess Techniques Of Finite Elment Methods

In this thesis,we mainly study superconvergence post-processing technique of finite element methods.In order to analysis the no-smooth problem,we give a new error estimate method by projection type interpolation and obtain extrapolation results of bilinear finite element with no-smooth solution.By more fine estiamte for high order Green function,we discuss the optimal estimate of recovery derivative about Quadratic Triangular Finite Element and odd-order rectangular finite element.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 high order discrete Green's function theory which be generalization of one order discrete Green's function.By fine estimate for high order discrete Green function,it is more convenient to study the optimal super-convergence of high order rectangular finite element.In Chapter 3,we introduce the theory of the interpolation operator of projection type and define a new error estimate order and it will be ease to analysis the supre-convergence and post-processing of no-smooth problem.In Chapter 4,By the new error estimate method given in chapter 3,we discuss the supconvergence of high order rectangular finite element.In Chapter 5,we discuss the superconvergence and extrapolation of bilinear finite element with no-smooth solution by the new error estimate method.In Chapter 6,By using the estimation of high order discrete Green function,we discuss the recovery derivative of quadratic triangular finite element and odd-order rectangular finite element and obtain the optimal estimation.