One-dimensional Finite Element Approximation Ultraconvergence

In this thesis, we study ultraconvergence of finite element approximation in one dimensional case (include two-point boundary value problem, one dimensional parabolic equation and parabolic type integro-differential equation ). Besides introduction and preliminary, there are three main parts.In the first part, we consider a class of two-point boundary value problems. We obtained interior ultraconvergent points for function and derivative using projection interpolation. These points depend on the solution u, but we can locate them by a simple formular with uh. Furthermore, we discuss SPR technique and finite element correction, we proved some results of SPR technique and obtained global ultraconvergence by correction.In the second part, we extend the above results of two-point boundary value problem to a class of parabolic equations. We obtained interior ultraconvergent points for function and derivative. We also obtained global ultraconvergence by correction.In the third part, for a class of parabolic integro-differential equations we obtained ultra-approximation between u and its Ritz-Volterra projection which leads to a ultraconvergence alternating theorem.