论文标题
一种用于非差异形式的二阶椭圆方程的修改的原始双重弱的绿素有限元法
A Modified Primal-Dual Weak Galerkin Finite Element Method for Second Order Elliptic Equations in Non-Divergence Form
论文作者
论文摘要
修改的原始双重弱甘蓝（M-PDWG）有限元方法的设计用于非差异形式的二阶椭圆方程。与\ cite {wwnondiv}中提出的现有PDWG方法相比，通过消除双重变量（lagrange multipleer），可以将M-PDWG方案产生的方程式等值简化为一个方程。因此，所得的简化系统的自由度明显少于现有PDWG方案所产生的自由度。此外，当适当选择M-PDWG方案中新引入的双线性术语时，可以大大减少简化系统的条件数。在离散的$ H^2 $ -NORM，$ H^1 $ -NORM和$ L^2 $ -NORM中的数值近似值中得出了最佳订单误差估计。对于凸面和非凸域上的平滑和非平滑系数证明了广泛的数值结果，以验证本文中开发的理论的准确性。
A modified primal-dual weak Galerkin (M-PDWG) finite element method is designed for the second order elliptic equation in non-divergence form. Compared with the existing PDWG methods proposed in \cite{wwnondiv}, the system of equations resulting from the M-PDWG scheme could be equivalently simplified into one equation involving only the primal variable by eliminating the dual variable (Lagrange multiplier). The resulting simplified system thus has significantly fewer degrees of freedom than the one resulting from existing PDWG scheme. In addition, the condition number of the simplified system could be greatly reduced when a newly introduced bilinear term in the M-PDWG scheme is appropriately chosen. Optimal order error estimates are derived for the numerical approximations in the discrete $H^2$-norm, $H^1$-norm and $L^2$-norm respectively. Extensive numerical results are demonstrated for both the smooth and non-smooth coefficients on convex and non-convex domains to verify the accuracy of the theory developed in this paper.
