一维Galerkin有限元EEP超收敛计算的加强格式  

ENHANCED FORM FOR EEP SUPER-CONVERGENCE CALCULATION IN ONE-DIMENSIONAL GALERKIN FINITE ELEMENT METHOD

作　　者：袁驷[1] 杨帅[1] YUAN Si;YANG Shuai(Key Laboratory of Civil Engineering Safety and Durability of China Education Ministry,Department of Civil Engineering of Tsinghua University,Beijing 100084,China)

机构地区：[1]清华大学土木工程系,土木工程安全与耐久教育部重点实验室,北京100084

出　　处：《工程力学》2025年第5期1-8,共8页Engineering Mechanics

基　　金：国家自然科学基金项目(51878383,51378293)。

摘　　要：基于对单元能量投影(element energy projection,EEP)法误差项的直接推导及分析,用EEP简约格式的解计算出略掉的误差项,反补后得到比简约格式高一阶精度的EEP超收敛计算的加强格式。该文以一维Galerkin有限元为例,给出EEP加强格式的算法公式和数学证明。理论分析和算例验证表明:对于m (≥1)次单元,采用EEP加强格式计算的内点位移和内点导数都具有h^(min(m+3,2m))阶的收敛精度,对系数特例问题二者甚至可以分别达到h^(min(m+5,2m))和h^(min(m+4,2m))阶的收敛精度。并对该法的进一步拓展作了讨论。Based on the direct derivation and analysis of the error terms of Element Energy Projection(EEP)method,the omitted error terms in the EEP solution of the simplified form are calculated,and the subsequent accuracy recovery results in an EEP enhanced form with convergence accuracy of one order higher than that of the EEP simplified form.Taking the one-dimensional Galerkin finite element as an example,the formulas and mathematical proofs of the EEP enhanced form are given.Theoretical analysis and numerical verification show that using elements with degree m≥1,the element interior displacements and derivatives calculated by the EEP enhanced form can both gain a convergence order of h^(min(m+3,2m)),and for problems with special coefficients,the convergence accuracy can even reach to the orders of h^(min(m+5,2m)) and h^(min(m+4,2m)).Potential further extension and development of this method are also addressed in this paper.

关 键 词：GALERKIN有限元 一维问题 超收敛 单元能量投影(EEP) 加强格式 

分 类 号：O342[理学—固体力学]