Postprocessed Two-Scale Finite Element Discretizations, Part I
Postprocessed Two-Scale Finite Element Discretizations, Part I
复制标题
DOI：
10.1137/11082292x
复制
发表时间：
2011-09
期刊：
影响因子：
--
通讯作者：
Fang Liu;M. Stynes;Aihui Zhou
中科院分区：
文献类型：
--
作者：
Fang Liu;M. Stynes;Aihui Zhou
本文研究了椭圆偏微分算子的二尺度有限元离散化的简单后处理，其中检查了张量积域 $\Omega\subset\mathbb{R}^d$ 上的边值问题和特征值问题，并使用了符合分段 $d$ 线性元素。对于$d\geq2$，假设使用粗网格和一些满足$H=\mathcal{O}(h^{2/3})$的单变量细网格应用二尺度方法，其中$h$和$H$是细网格和粗网格宽度。对于这两类问题，结果表明，后处理的双尺度有限元解在 $H^1(\Omega)$ 范数中与后处理的标准有限元解达到了相同数量级的精度，但与后者所需的 $\mathcal{O}(h^{-d})$ 自由度相比，前一个解仅使用 $\mathcal{O}(h^{-(2d+1)/3})$ 自由度。
This paper studies a simple postprocessing of two-scale finite element discretizations of elliptic partial differential operators, where boundary value problems and eigenvalue problems on tensor-product domains $\Omega\subset\mathbb{R}^d$ are both examined and conforming piecewise $d$-linear elements are used. For $d\geq2$, suppose that the two-scale method is applied using a coarse mesh and some univariate fine meshes satisfying $H=\mathcal{O}(h^{2/3})$, where $h$ and $H$ are the fine and coarse mesh widths. It is shown for both classes of problem that the postprocessed two-scale finite element solution achieves the same order of accuracy in the $H^1(\Omega)$ norm as the postprocessed standard finite element solution, but the former solution uses only $\mathcal{O}(h^{-(2d+1)/3})$ degrees of freedom compared with the $\mathcal{O}(h^{-d})$ degrees of freedom required by the latter.