Superconvergence of High Order Finite Difference Schemes Based on Variational Formulation for Elliptic Equations
Superconvergence of High Order Finite Difference Schemes Based on Variational Formulation for Elliptic Equations
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DOI：
10.1007/s10915-020-01144-w
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发表时间：
2020-02-01
影响因子：
2.5
通讯作者：
Zhang, Xiangxiong
中科院分区：
文献类型：
--
作者：
Li, Hao;Zhang, Xiangxiong
当用(k+1)x(k+1)Gauss-Lobatto积分代替所有积分时，经典的拉格朗日QK基连续有限元方法简化为有限差分格式。对于具有Dirichlet边界条件的椭圆型方程，我们证明了该格式在离散2-范数意义下是(k+2)阶精度的，这是函数值超收敛的结果。对于k=2的情形，我们还给出了一个简单的四阶精确椭圆求解器在矩形区域上的方便实现。
The classical continuous finite element method with Lagrangian Qk basis reduces to a finite difference scheme when all the integrals are replaced by the (k+1)x(k+1)Gauss-Lobatto quadrature. We prove that this finite difference scheme is (k+2) order accurate in the discrete 2-norm for an elliptic equation with Dirichlet boundary conditions, which is a superconvergence result of function values. We also give a convenient implementation for the case k=2, which is a simple fourth order accurate elliptic solver on a rectangular domain.