抛物型积分微分方程各向异性非协调有限元分析  

Anisotropic Nonconforming Finite Element Analysis for Integro-differential Equations of Parabolic Type

作　　者：石东洋[1] 王海红[1] 

机构地区：[1]郑州大学数学系,郑州450052

出　　处：《工程数学学报》2009年第2期209-218,共10页Chinese Journal of Engineering Mathematics

基　　金：国家自然科学基金(10671184)

摘　　要：各向异性有限元方法的显著的优点之一就是可以用较少的自由度得到与传统有限元正则剖分时同样的估计结果。然而,在这种情况下,Sobolev空间上的Bramble-Hilbert引理在插值误差分析中不能直接应用,而且对于非协调元来说其传统边界估计技巧也不再适用。本文证明了一个非协调单元具有各向异性特征,并将它应用到研究抛物积分微分方程半离散格式下的Galerkin逼近。利用单元的特殊性,验证了Ritz-Volterra投影与有限元插值是相同的。在解适当光滑时,通过引入一些新的技巧,得到了与传统方法相同的收敛误差估计和超逼近性质。最后,通过构造适当的插值后处理算子,得到了各向异性网格下的整体超收敛结果。该文的结果对进一步探索和设计数值的自适应算法是有帮助的。One of the outstanding advantages about anisotropic finite element methods is to allow us to achieve the same accuracy estimates with less degrees of freedom as compared with traditional finite element methods with regular subdivisions. However, in this case, the Bramble-Hilbert lemma in Sobolev space can not be directly used in the interpolation error analysis and usual boundary estimate techniques are not applicable to nonconforming finite elements. In this paper, we prove that a kind of nonconforming element has anisotropic characteristic and apply it to the Galerkin approximation of parabolic type integro-differential equations under semi-discrete scheme. By use of the special property of the element, we also prove that the Ritz-Volterra projection is the same as the finite element interpolation. By introducing some new techniques, the same convergence error estimates and supperclose property as conventional method are obtained when the solution is appropriately smooth. Finally, the global superconvergence result under anisotropic meshes is obtained by constructing a proper interpolation post-processing operator. The results of this paper are helpful for further exploring and designing self-adapting algorithm about numerical solutions.

关 键 词：抛物型积分微分方程 各向异性 非协调元 超逼近及超收敛 

分 类 号：O242.21[理学—计算数学]