四阶奇异摄动问题的弱Galerkin有限元法  

Weak Galerkin finite element method for fourth order singular perturbation problems

作　　者：王琳 罗鲲[1] 张世全[1] WANG Lin;LUO Nun;ZHANG Shi-Quan(School of Mathematics,Sichuan University,Chengdu 610064,China)

机构地区：[1]四川大学数学学院,成都610064

出　　处：《四川大学学报（自然科学版）》2018年第6期1141-1147,共7页Journal of Sichuan University(Natural Science Edition)

基　　金：国家自然科学基金(11401407)

摘　　要：本文研究二维和三维情形下四阶奇异摄动问题弱Galerkin有限元法的构造与分析.我们引入了弱二阶偏导数算子,对单元内部的位移变量采用连续分片k(k≥2)次多项式逼近,对单元边界上的位移梯度采用间断分片k-1次多项式逼近.基于Scott-Zhang和L2投影算子的性质,该方法能够得到能量范数的最优误差估计,且针对边界层问题,能够得到与摄动参数一致无关的收敛阶.数值算例验证了理论结果.In this paper,we discuss the construction and analysis of the weak Galerkin （WG） finite element method for the fourth order singular perturbation problems in two and three dimensions. By introducing the weak second order partial derivative operators,the WG method is constructed by adopting continuous piecewise polynomials of degree k（k≥2） for the approximation to the displacement in the interior of elements,and discontinuous piecewise polynomials of degree k-1 for the approximations to the trace of displacement gradient on the inter-element boundaries. Based on the properties of the Scott-Zhang and L 2 projections,optimal error estimates in energy norm are derived. In addition,for the boundary layer case,we show that the methods are convergent uniformly with respect to the perturbation parameter. Numerical examples are provided to verify the theoretical results.

关 键 词：弱Galerkin有限元方法 四阶奇异摄动问题 边界层 

分 类 号：O241.82[理学—计算数学]