超罚弱有限元方法求解二阶椭圆问题
发布时间：2018-11-05 14:42
【摘要】：弱有限元(Weak Galerkin,简称WG)方法首先是由王军平和叶秀等人提出利用弱函数和弱梯度来求解二阶椭圆问题.弱函数空间的选取依赖于定义在网格单元的内部和边界上的多项式空间,这使得弱有限元方法在许多应用上更加的灵活可靠.一般来说,间断有限元(Discontinuous Galerkin)方法定义的跳跃来自分片单元内部函数在单元边界上函数值;弱有限元方法传统做法是在单元边界上定义单值函数,与内部单元函数利用弱函数定义联系.本文与之区别在于,我们在单元边界上定义双值函数,于是在同一个单元边界上就自然产生两个弱函数的差,我们称之为弱跳跃.基于内罚间断Galerkin有限元的思想,我们对单元边界上的弱跳跃加罚项,就形成本文介绍的超罚弱有限元方法.本文主要以具有光滑解的二阶椭圆问题为例讨论超罚弱有限元方法.文中首先给出了二阶椭圆问题的数值格式,严格证明了在H1-范数和L2-范数意义下基于弱函数空间(Pk,Pk,RTk)(k≥0)的先验误差估计,并且给出相关的数值实验来验证理论结果.本文还给出另外一种同时带有超罚项和稳定项的WG方法,它基于弱函数空间(Pk,Pk,[Pk-1]2)(k≥1)或者(Pk,Pk-1,[Pk-1]2)(k≥1).该算法格式中网格剖分单元不再局限于单纯形,而是可以扩展到一般多边形或多面体,并且理论和实验都表明,通过选取适当的罚参数同样可以达到最优收敛阶.从算法的格式和数值实现来看,它是弱有限元方法的一种自然拓广,具有逼近函数简单,网格生成灵活,并且其数值格式绝对稳定,单元刚度矩阵可以独立实现,便于并行计算等优点.值得一提的是,本文虽然只给出了二阶椭圆问题的超罚弱有限元格式,但是可以将这个方法应用到其它常见的偏微分方程上,如椭圆界面问题,Stokes方程,div-curl系统等.
[Abstract]:The weak finite element (Weak Galerkin, (WG) method is firstly proposed by Wang Junping and Ye Xiu to solve the second order elliptic problem by using weak function and weak gradient. The selection of the weak function space depends on the polynomial space defined on the interior and boundary of the grid element, which makes the weak finite element method more flexible and reliable in many applications. In general, the jump defined by the discontinuous finite element (Discontinuous Galerkin) method comes from the function value of the internal function of the piecewise element on the boundary of the element. The traditional method of weak finite element method is to define the single-valued function on the boundary of the element, which is related to the definition of the internal element function by using the weak function. The difference between this paper and the other is that we define the two-valued function on the element boundary, so the difference between the two weak functions on the same cell boundary is naturally generated, which is called weak jump. Based on the idea of discontinuous Galerkin finite element with internal penalty, we add penalty term to the weak jump on the boundary of the element, and form the super-penalty weak finite element method introduced in this paper. In this paper, the second order elliptic problem with smooth solution is taken as an example to discuss the superpenalty weak finite element method. In this paper, the numerical scheme of the second order elliptic problem is given, and a priori error estimate based on weak function space (Pk,RTk) (k 鈮，
本文编号：2312392
[Abstract]:The weak finite element (Weak Galerkin, (WG) method is firstly proposed by Wang Junping and Ye Xiu to solve the second order elliptic problem by using weak function and weak gradient. The selection of the weak function space depends on the polynomial space defined on the interior and boundary of the grid element, which makes the weak finite element method more flexible and reliable in many applications. In general, the jump defined by the discontinuous finite element (Discontinuous Galerkin) method comes from the function value of the internal function of the piecewise element on the boundary of the element. The traditional method of weak finite element method is to define the single-valued function on the boundary of the element, which is related to the definition of the internal element function by using the weak function. The difference between this paper and the other is that we define the two-valued function on the element boundary, so the difference between the two weak functions on the same cell boundary is naturally generated, which is called weak jump. Based on the idea of discontinuous Galerkin finite element with internal penalty, we add penalty term to the weak jump on the boundary of the element, and form the super-penalty weak finite element method introduced in this paper. In this paper, the second order elliptic problem with smooth solution is taken as an example to discuss the superpenalty weak finite element method. In this paper, the numerical scheme of the second order elliptic problem is given, and a priori error estimate based on weak function space (Pk,RTk) (k 鈮，
本文编号：2312392
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