二维泊松方程问题Lagrange型有限元p型超收敛算法  

A p-TYPE SUPERCONVERGENT RECOVERY METHOD FOR FE ANALYSIS WITH LAGRANGE ELEMENTS ON TWODIMENSIONAL POISSON EQUATIONS

作　　者：叶康生[1] 孟令宁 YE Kang-sheng;MENG Ling-ning(Department of Civil Engineering,Tsinghua University,Key Laboratory of Civil Engineering Safety and Durability of China Education Ministry,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》2022年第2期23-36,共14页Engineering Mechanics

基　　金：国家自然科学基金项目(51078198);清华大学自主科研计划项目(2011THZ03)。

摘　　要：该文针对二维泊松方程问题的Lagrange型有限元法提出了一种p型超收敛算法。该法受有限元线法对二维问题降维思想的启发,基于网格结点位移的天然超收敛性,通过从网格中取出一行对边相邻的单元作一子域,将子域内各单元另一对边解答取为原有限元解答,在子域上建立真解近似满足的局部偏微分方程边值问题,对该局部边值问题,沿对边方向单向提高单元阶次进行有限元求解获得单元对边上的超收敛解。单元另一对边上的超收敛解可通过另一方向的单元行类似获得。在单元边超收敛解的基础上,依次取出各个单元,以单元边位移超收敛解为Dirichlet边界条件,双向提高单元阶次对原泊松方程问题进行有限元求解即可获得全域超收敛解。数值算例表明,通过简单的后处理计算本法可显著提高解答的精度和收敛阶。A p-type superconvergent recovery method for the finite element analysis with Lagrange elements on two-dimensional Poisson equations is proposed.Based on the superconvergent properties of mesh nodal displacements in finite element solutions,the method is inspired by the idea of dimensionality reduction of two-dimensional problems by the finite element method of lines.A row of adjacent elements with common opposite edges is taken out as a sub-domain.A local boundary value problem of the original partial differential equations on it which the true solution approximately satisfies is established by setting finite element solutions on each element's other opposite edges as Dirichlet boundary conditions.By increasing the element order along the elements'opposite edges direction unidirectionally,the local boundary value problem is solved by the finite element method to obtain the superconvergent displacement solution on the opposite edges of each element in this sub-domain.The superconvergent solution on the other opposite edges of elements can be obtained similarly with another sub-domain with respect to the edges to be recovered.Based on the recovered edge solutions,each element domain is taken out.The original Poisson equation on it is solved using a higher order Lagrange element with the superconvergent solution on its edges set as Dirichlet boundary conditions.Thus,the superconvergent solution of the whole domain can be obtained.Numerical examples show that the method can significantly improve the accuracy and convergence order of solutions with a small amount of computation.

关 键 词：有限元 p型超收敛 后处理 泊松方程 边值问题 

分 类 号：O242.21[理学—计算数学]