Superconvergence of solution derivatives for the Shortley-Weller difference approximation of Poisson's equation. Part I: smoothness problems
Superconvergence of solution derivatives for the Shortley-Weller difference approximation of Poisson's equation. Part I: smoothness problems
复制标题
泊松方程的 Shortley-Weller 差分近似解导数的超收敛。
DOI：
10.1016/s0377-0427(02)00754-9
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发表时间：
2003
影响因子：
2.4
通讯作者：
Qing Fang
中科院分区：
文献类型：
--
作者：
Zi;Tetsuro Yamamoto;Qing Fang
采用Shortley-Weller近似的有限差分法可以看作是一种特殊的有限元方法，它使用分段双线性和线性函数，并涉及到一定的积分逼近。当u∈C3(S̄)(即u∈C3，0(S̄))和f∈C2(S̄)时，对于矩形差分网格，导出了有限差分法求解离散H1范数导数的超收敛速度O(H2)，其中h是所用的差分网格的最大网格长度，且差分网格不限于准均匀。比较了极值原理分析和有限元分析，讨论了有限差分法与线性和双线性有限元分析之间的转换，并给出了数值实验以支持超收敛分析。
The finite difference method (FDM) using the Shortley–Weller approximation can be viewed as a special kind of the finite element methods (FEMs) using the piecewise bilinear and linear functions, and involving some integration approximation. When u∈C3( S ̄ ) (i.e., u∈C3,0( S ̄ ) ) and f∈C2( S ̄ ) , the superconvergence rate O(h2) of solution derivatives in discrete H1norms by the FDM is derived for rectangular difference grids, where h is the maximal mesh length of difference grids used, and the difference grids are not confined to be quasiuniform. Comparisons are made on the analysis by the maximum principle and the FEM analysis, conversions between the FDM and the linear and bilinear FEMs are discussed, and numerical experiments are provided to support superconvergence analysis made.