Finite element superconvergence on Shishkin mesh for 2-D convection-diffusion problems
Finite element superconvergence on Shishkin mesh for 2-D convection-diffusion problems
复制标题
DOI：
10.1090/s0025-5718-03-01486-8
复制
发表时间：
2003-07
期刊：
影响因子：
--
通讯作者：
Zhimin Zhang
中科院分区：
文献类型：
--
作者：
Zhimin Zhang
本文研究了二维对流扩散问题的Shishkin网格双线性有限元方法。在一定的正则性假设下，得到了离散e-权能量范数下的超收敛速度O（N-2ln 2N + eN-1.5lnN）.这个收敛速度对于奇异摄动参数e是一致有效的。数值试验表明，边界层项的变化率为O（N-2ln 2N）。作为副产品，在L2-范数下得到了相同阶的e-一致收敛性。在同样的正则性假设下，证明了边界层区域内某些网格点在L∞范数下具有N-3/2ln 5/2N + eN-1 ln 1/2N阶的e-一致收敛性.
In this work, the bilinear finite element method on a Shishkin mesh for convection-diffusion problems is analyzed in the two-dimensional setting. A superconvergence rate O(N-2 ln2 N + eN-1.5lnN) in a discrete e-weighted energy norm is established under certain regularity assumptions. This convergence rate is uniformly valid with respect to the singular perturbation parameter e. Numerical tests indicate that the rate O(N-2ln2 N) is sharp for the boundary layer terms. As a by-product, an e-uniform convergence of the same order is obtained for the L2-norm. Furthermore, under the same regularity assumption, an e-uniform convergence of order N-3/2ln5/2N + eN-1ln1/2N in the L∞ norm is proved for some mesh points in the boundary layer region.