Asymptotic expansions of finite element solutions to Robin problems in H^3 and their application in extrapolation cascadic multigrid method  

作　　者：HU HongLing CHEN ChuanMiao PAN KeJia 

机构地区：[1]College of Mathematics and Computer Science,Key Laboratory of High Performance Computing and Stochastic Information Processing (Ministry of Education of China),Hunan Normal University [2]School of Mathematics and Statistics,Central South University [3]Key Laboratory of Metallogenic Prediction of Nonferrous Metals,Ministry of Education,School of Geosciences and Info-Physics,Central South University

出　　处：《Science China Mathematics》2014年第4期687-698,共12页中国科学：数学（英文版）

基　　金：supported by National Natural Science Foundation of China(Grant Nos.11226332,41204082 and 11071067);the China Postdoctoral Science Foundation(Grant No.2011M501295);the Research Fund for the Doctoral Program of Higher Education of China(Grant No.20120162120036);the Construct Program of the Key Discipline in Hunan Province

摘　　要：For the Poisson equation with Robin boundary conditions,by using a few techniques such as orthogonal expansion(M-type),separation of the main part and the finite element projection,we prove for the first time that the asymptotic error expansions of bilinear finite element have the accuracy of O(h3)for u∈H3.Based on the obtained asymptotic error expansions for linear finite elements,extrapolation cascadic multigrid method(EXCMG)can be used to solve Robin problems effectively.Furthermore,by virtue of Richardson not only the accuracy of the approximation is improved,but also a posteriori error estimation is obtained.Finally,some numerical experiments that confirm the theoretical analysis are presented.For the Poisson equation with Robin boundary conditions,by using a few techniques such as orthogonal expansion（M-type）,separation of the main part and the finite element projection,we prove for the first time that the asymptotic error expansions of bilinear finite element have the accuracy of O（h3）for u∈H3.Based on the obtained asymptotic error expansions for linear finite elements,extrapolation cascadic multigrid method（EXCMG）can be used to solve Robin problems effectively.Furthermore,by virtue of Richardson not only the accuracy of the approximation is improved,but also a posteriori error estimation is obtained.Finally,some numerical experiments that confirm the theoretical analysis are presented.

关 键 词：finite element Richardson extrapolation Robin problem asymptotic expansion cascadic multi-grid method 

分 类 号：O241.82[理学—计算数学] O175.14[理学—数学]