3维奇异摄动四阶旋度问题的非协调有限元逼近及其Nitsche方法分析  

Nonconforming finite element approximations and the analysis of Nitsche’s method for a singularly perturbed quad-curl problem in three dimensions

作　　者：张百驹 张智民 Baiju Zhang;Zhimin Zhang

机构地区：[1]云南大学数学与统计学院,昆明650500 [2]Department of Mathematics,Wayne State University,Detroit,MI 48202,USA

出　　处：《中国科学：数学》2024年第4期647-670,共24页Scientia Sinica：Mathematica

基　　金：国家自然科学基金(批准号:12131005)资助项目。

摘　　要：本文对3维奇异摄动四阶旋度模型介绍并分析了一种稳健的非协调有限元方法.对于模型问题的解,本文给出了相应的先验估计,并基于此证明了所提出的有限元方法关于奇异摄动参数ε是稳健的,还证明了数值解以h^(1/2)一致收敛.此外,本文还探索了利用Nitsche方法弱处理第二边界条件的效果,并证明了,当ε<h时,与强加边界条件的情形相比,这样的处理能得到更好的误差估计.最后,数值实验验证了该方法的良好性能并证实了本文的理论预测.We introduce and analyze a robust nonconforming finite element method for a three-dimensional singularly perturbed quad-curl model problem.For the solution of the model problem,we derive proper a priori bounds,based on which we prove that the proposed finite element method is robust with respect to the singular perturbation parameterεand the numerical solution is uniformly convergent with order h^(1/2).In addition,we investigate the effect of treating the second boundary condition weakly by Nitsche’s method.We show that such a treatment leads to sharper error estimates than imposing the boundary condition strongly when the parameterε<h.Finally,numerical experiments are provided to illustrate the good performance of the method and confirm our theoretical predictions.

关 键 词：四阶旋度问题 奇异摄动 非协调有限元 Nitsche方法 

分 类 号：O241.82[理学—计算数学]