论文标题
带有可变系数和vonKármán方程的Biharmonic方程的混合有限元方案
A mixed finite element scheme for biharmonic equation with variable coefficient and von Kármán equations
论文作者
论文摘要
在本文中，为Biharmonic方程引入了一种新的混合有限元方案，该方程在Lipschitz多面体域上具有可变系数。所提出的方案不涉及沿网格接口的任何集成。该解决方案的梯度由$ h（{\ rm div}）$ - 符合$ bdm_ {k+1} $元素或带有$ k+1 $订单的vector的lagrange元素，而该解决方案近似于lagrange元素与$ k+2 $的$ k+2 $ for $ k \ egeq 0 $ .q \ egeq 0 $。我们提供了一个新的离散$ H^{2} $ - 标准稳定性，它不仅在对该方案的分析中，而且在$ C^{0} $内部惩罚方法和DG方法中也很有用。派生的$ h^{2} $ - norm和$ l^{2} $ - norm中的最佳收敛均被派生。该方案及其分析进一步推广到vonKármán方程。最后，还提出了验证所提出算法的理论估计值的数值结果。
In this paper, a new mixed finite element scheme using element-wise stabilization is introduced for the biharmonic equation with variable coefficient on Lipschitz polyhedral domains. The proposed scheme doesn't involve any integration along mesh interfaces. The gradient of the solution is approximated by $H({\rm div})$-conforming $BDM_{k+1}$ element or vector valued Lagrange element with order $k+1$, while the solution is approximated by Lagrange element with order $k+2$ for any $k\geq 0$.This scheme can be easily implemented and produces positive definite linear system. We provide a new discrete $H^{2}$-norm stability, which is useful not only in analysis of this scheme but also in $C^{0}$ interior penalty methods and DG methods. Optimal convergences in both discrete $H^{2}$-norm and $L^{2}$-norm are derived. This scheme with its analysis is further generalized to the von Kármán equations. Finally, numerical results verifying the theoretical estimates of the proposed algorithms are also presented.
