具有最佳超收敛阶的EEP法计算格式:Ⅲ数学证明

A SCHEME WITH OPTIMAL ORDER OF SUPER-CONVERGENCE BASED ON EEP METHOD: III MATHEMATICAL PROOF

对一维C0问题的高次有限元后处理中超收敛计算的EEP(单元能量投影)法提出改进的最佳超收敛计算格式,即用m次单元对足够光滑问题的有限元解答,采用该格式计算的任一点的位移和应力都可以达到h2m阶的最佳超收敛结果。整个工作分为3个部分,分别给出算法公式、数值算例和数学证明。该文是系列工作的第三部分,对所提出的最佳的EEP超收敛格式给出数学证明。

Based on the Element Energy Projection （EEP） method, an improved scheme with optimal order of super-convergence, is presented for one-dimensional C^0 FEM, i.e., FEM sulotions can be obtained through the scheme for the elements with sufficient smooth property and m degrees. The proposed scheme is capable of producing O（h^2m） super-convergence for both displacements and stresses at any point on an element in postprocessing stage. The entire work is composed of three parts, i.e. formulation, numerical results as well as mathematical analysis. The present paper is the third in the series and gives the mathematical proof of the optimal O（h^2m） super-convergence for the proposed scheme.