有限元超收敛新论

本文从三个方面讨论二阶椭圆问题有限元超收敛．1．一致网格上的新超收敛结果．利用新的“投影型插值”,我们解决了高次三角形元的超收敛问题．2．一般网格的超收敛性．利用局部插值处理和局部磨光处理我们获得了整体超收敛性结果．3．关于当前的两种超收敛技巧．Cornell学派利用一个精致的内估计和网格的点对称性,获得了一个“普遍”的结果,中国学派利用两个基本估计和离散Green函数理论获得了令人满意的结果,两者均很复杂．本文综合了两个学派的方法,简洁地证得上述普遍结果．

New Discussions for Finite Element superconvergence

This paper discusses superconvergence for 2nd order elliptic problems in three main aspects.1. New superconvergence results in uniform meshes. We solved the superconvergence for higher order triangular elements using a new "projection interpolation".2. Superconvergence of free meshes. Using local interpolation processing and local smoothing processing we obtained global superconvergence.3. On the two current superconvergence techniques. Cornell school using a elaborative interior estimate and point symmetric meshes obtained very general results. China school using two fundamental estimates and discrete Green's function theory obtained satisfied results. Both are too complicate to understand. This paper using a synthesis method obtained a general result. The proof is simple and clear.