The Superconvergence Postprocess Techniques Of Finite Element Methods

The finite element method is one of the efficient numerical methods to solve partial different equations(PDEs). But the derivatives of the finite element solution do not continue over elements' boundary and have low global accuracy. So it arises many computation mathematicians' interest how to improve the accuracy of the finite element solutions.In 1992 Zienkiewicz and Zhu proposed the superconvergence patch re-covery, i.e. the SPR technique. Because of its advantage, such as its simplicity for calculation, easiness to understand, convenience to access to the application software for the finite element method etc., this technique has been generally applied by the engineering community, and is thought of, by Babuska etal., as one of the most[3]robust posterior error estimators which are asymptotically exact . Since that the postprocess technique of finite element method had gone from theoretical study to realistically application. It is of interest to note that the ultraconvergence, i.e. with two order higher than the global optimal convergent rate, of the derivatives at the points of interest is obtained by the SPR technique on the uniform meshes in either one or two space dimensions for all even-order elements. But it is a pity that only superconvergence, not ultraconvergence, is achieved by this method for odd-order finite elements.Since 1995 Z. M. Zhang, J. Z. Zhu, B. Li etal. have respectively proved the superconvergent results for two-point boundary value problems, rectangular finite elements and linear triangular finite element by SPR technique. But the ultracon-vergent results for quadratic triangular finite element has been challenging, and has not been justified.In this paper the superconvergence postprocess technique of FEM is discussed. The mending of SPR technique is discussed based on the asymptotic expansion of FEM, and the ultraconvergence of derivatives for odd-degree finite elements is obtained, which makes up the limitation of SPR technique;the difficult problem for quadratic triangular finite elements is proved base on studying the properties of the recovery operators and by symmetric technique.In this paper many results are achieved, the most innovation is1. A new recovery technique is proposed for odd-degree finite elements in one and two space dimensions, and the ultraconvergence of derivatives, even the O(h6) results for cubic finite elements, is obtained by this method, which can not be obtained by the SPR technique and makes up the limitation of SPR. cf. Chapter 5§5-2;2. We mend the SPR technique for quadratic finite elements, and obtain the same ultraconvergent results as SPR. And we prove the results by SPR and our method, which has not been proved by anyone, cf. Chapter 5 §5.1.3. A new idea to prove the superconvergence of recovery operators is presented based on super-approximation of FEM. cf. Chapter 5;4. The very particular estimations of projection type interpolation is presented, which plays an important role in superconvergence theoretical analysis, particularly for singular problems, cf. Chapter 2;5. We expand the bilinear type a(u - ipu, v) for bi-cubic finite elements, and obtain a good ultra-approximation results, cf. Chapter 3 §3.3.