Convergence analysis of the adaptive finite element method with the red-green refinement  

作　　者：ZHAO XuYing 1, 3 , HU Jun 2,? & SHI ZhongCi 1 1 State Key Laboratory of Scientific and Engineering Computing, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China 2 Key Laboratory of Mathematics and Applied Mathematics (Peking University), Ministry of Education and School of Mathematical Sciences, Peking University, Beijing 100871, China 3 Graduate University of Chinese Academy of Sciences, Beijing 100190, China 

出　　处：《Science China Mathematics》2010年第2期499-512,共14页中国科学：数学（英文版）

基　　金：supported by National Natural Science Foundation of China (Grant Nos.10601003, 10971005);Foundation for the Author of National Excellent Doctoral Dissertation of China (Grant No.200718) ;National Basic Research Program of China (Grant No. 2005CB321704)

摘　　要：In this paper, we analyze the convergence of the adaptive conforming P 1 element method with the red-green refinement. Since the mesh after refining is not nested into the one before, the Galerkin-orthogonality does not hold for this case. To overcome such a difficulty, we prove some quasi-orthogonality instead under some mild condition on the initial mesh (Condition A). Consequently, we show convergence of the adaptive method by establishing the reduction of some total error. To weaken the condition on the initial mesh, we propose a modified red-green refinement and prove the convergence of the associated adaptive method under a much weaker condition on the initial mesh (Condition B).In this paper, we analyze the convergence of the adaptive conforming P 1 element method with the red-green refinement. Since the mesh after refining is not nested into the one before, the Galerkin-orthogonality does not hold for this case. To overcome such a difficulty, we prove some quasi-orthogonality instead under some mild condition on the initial mesh (Condition A). Consequently, we show convergence of the adaptive method by establishing the reduction of some total error. To weaken the condition on the initial mesh, we propose a modified red-green refinement and prove the convergence of the associated adaptive method under a much weaker condition on the initial mesh (Condition B).

关 键 词：red-green REFINEMENT adaptive finite element method convergence non-nested local REFINEMENT 

分 类 号：O241.82[理学—计算数学]