有限元网格的超收敛测度及其应用

Superconvergence Measurement for General Meshes by Linear Finite Element Method

给出线性有限元求解二阶椭圆问题的有限元网格超收敛测度及其应用.有限元超收敛经常是在具有一定结构的特殊网格条件下讨论的,而本文从一般网格出发,导出一种网格的范数用来描述超收敛所需要的网格条件以及超收敛的程度.并且通过对这种网格范数性质的考察,可以证明对于通常考虑的一些特殊网格的超收敛的存在性.更进一步,我们可以通过正则细分的方式在一般区域上也可以自动获得超收敛网格.最后给出相关的数值结果来验证本文的理论分析.

In this paper,we derive a type of superconvergence measurement of triangular meshes for the second order elliptic problems by the linear finite element method.We know that finite element superconvergence always needs some kinds of structured meshes.Here, we define a mesh norm to measure the superconvergence for triangular meshes.Based on the property investigation of the superconvergence measurement,we can prove there exists superconvergence for the classical structured meshes.Furthermore,we can construct suitable superconvergence meshes on the general computing domains.Some numerical results are given to validate our theoretical analysis.