矩形双线性元在弱空间中的整体高精度分析及应用

Superconvergence and extrapolation is one of the most effective means to improve the accuracy in modern scientific and engineering computing..In the past when we study the high-precision method in the sense of maximum norm, we must remove some small area for non-uniform grid. So maximum-norm is not an overall norm and it does not suitable for EXCMG. Changing the normal way of thinking and avoiding Green''s function, the project will study in the sense of discrete L^2-norm that is weak but suitable for the overall measure. And then with a few technique : duality argument, orthogonal expansion (M-type), separation of the main department and so on the project will deduce the error asymptotic expansion and extrapolation formulas of finite element solution. At last the results will be applied to the efficient algorithm——The New Extraplation Cascadic Multigrid Method(EXCMG). The project focus on studying on the high-precision results of rectangular bilinear finite element for elliptic problem with Dirichlet or Robin condition in H^3.

超收敛和外推是现代科学和工程计算中提高精度的有效手段之一。以往在最大模意义下研究高精度方法，对非均匀网格必须除去某些小区域，因此最大模不是一个整体范数，不适用于外推多网格法。本项目改变思维模式，在较弱的整体度量的离散L^2范数下进行研究，利用对偶论证，避开复杂的Green函数估计，借助正交展开和误差主部分离等技巧，得到整体的误差渐近展开式及外推公式。这些高精度结果可直接应用到最近发展起来的高效算法——外推瀑布式多网格法（EXCMG）。本项目重点在弱正则性H^3空间中研究椭圆第二、第三边值问题矩形双线性有限元的高精度结果及其在EXCMG算法中的应用。

本项目改变思维模式，在较弱的整体度量的离散L2范数下进行研究，利用对偶论证，避开复杂的Green函数估计，借助正交展开和误差主部分离等技巧，重点研究了弱正则性H3空间中的Robin边值问题矩形双线性有限元的高精度结果，得到整体的误差渐近展开式和外推公式。及其在外推瀑布式多网格法(EXCMG)中的应用。并研究了分块几乎均匀网格上的三角形线性元在H1范数意义下的整体超收敛性和离散L2范数意义下的渐近展开式和外推公式及在EXCMG的应用。且对于共轭梯度法(CG)的l2收敛性及基于CG的外推瀑布式多网格法在A范数意义下的收敛性都进行了讨论。