LOWEST-ORDER WEAK GALERKIN FINITE ELEMENT METHOD FOR DARCY FLOW ON CONVEX POLYGONAL MESHES
LOWEST-ORDER WEAK GALERKIN FINITE ELEMENT METHOD FOR DARCY FLOW ON CONVEX POLYGONAL MESHES
复制标题
DOI：
10.1137/17m1145677
复制
发表时间：
2018-01-01
影响因子：
3.1
通讯作者：
Wang, Zhuoran
中科院分区：
文献类型：
--
作者：
Liu, Jiangguo;Tavener, Simon;Wang, Zhuoran
在一般凸多边形网格上提出了求解Darcy方程或椭圆边值问题的最低阶弱Galerkin（WG）有限元方法。在这种方法中，常数被用于元素内部和边缘，以近似原始变量（压力）。这些常数基函数的离散弱梯度建立在多边形上的简单H（div）-子空间中，该多边形通过使用归一化坐标和Wachspress坐标显式构造[W. Chen和Y. Wang，Math. Comp.，86（2017），pp. 2053-2087年]。这些离散的弱梯度被用来近似的经典梯度的变分制剂。这种新方法不需要惩罚。该方法的结果在对称正定稀疏线性系统。它是局部质量保守的，并产生连续的法向通量。当凸多边形网格为规则网格时，新方法在压力、速度和法向流量方面具有最优阶收敛性。
This paper presents the lowest-order weak Galerkin (WG) finite element method for solving the Darcy equation or elliptic boundary value problems on general convex polygonal meshes. In this approach, constants are used in element interiors and on edges to approximate the primal variable (pressure). The discrete weak gradients of these constant basis functions are established in simple H(div)-subspaces on polygons that are explicitly constructed by using the normalized coordinates and Wachspress coordinates [W. Chen and Y. Wang, Math. Comp., 86 (2017), pp. 2053-2087]. These discrete weak gradients are used to approximate the classical gradient in the variational formulation. No penalization is needed for this new method. The method results in symmetric positive-definite sparse linear systems. It is locally mass-conservative and produces continuous normal fluxes. The new method has optimal-order convergence in pressure, velocity, and normal flux, when the convex polygon meshes are shape-regular.