STO KES问题的优化结果

高少芹

(河北大学数学与计算机学院,河北保定 071002)

研究报告

STO KES问题的优化结果

高少芹

(河北大学数学与计算机学院,河北保定 071002)

为了研究Stokes问题的最小二乘有限元近似解的超收敛结果,利用最小二乘曲面拟合的方法给出了Stokes问题的最小二乘有限元解中速度的优化结果.该结果表明:随着构成拟合空间的分片多项式次数的增加,理论上得到的速度的近似解精度越高.该结果是以Stokes问题的正则性为前提的.

Stokes问题;最小二乘有限元;速度;优化

1 预备知识

近年来,最小二乘有限元方法越来越被关注,可以用于求解数学、物理问题[1-4].而有限元解的超收敛也是一个有趣且在工业问题的科学计算中非常有用的问题[5-10],但有关最小二乘有限元解的超收敛方面的文章较罕见.文献[1]给出了Stokes问题的最小二乘有限元近似解中速度u的近似值uh的L2-范数误差估计,本文将给出速度u的一种优化的近似解,这种结果是利用文献[6-7]中对于标准Galerkin方法提出并作出分析的最小二乘曲面拟合方法得到的.

考虑Stokes问题

2 主要结果

3 结论

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[2]GAO Shaoqin.Least-squares mixed finite element methods for the incomp ressible magnetohyd rodynam ic Equations[J].Journal of Computational Mathematics,2005,23(3):327-336.

[3]BOCHEV PB,GUNZBURGER M D.Finite elementmethodsof least-squares type[J].SIAM Rev,1998,40:789-837.

[4]CA IZhiqiang,MANTEUFFEL T,MCCORM ICK S.First-o rder system least squares fo r the Stokes equations,w ith app lication to linear elasticity[J].SIAM J Numer Anal,1997,34(5):1727-1741.

[5]XIE Hehu,GAO Shaoqin.Superconvergence of the least-squaresmixed finite element app roximations fo r the second o rder elliptic p rop lem s[J].International Journal of Info rmation and System s Sciences,2005,1(1):1-6.

[6]WANG Junping.A supperconvergence analysis fo r finite element solutions by the least-squares surface fitting on irregular meshes fo r smooth p roblem s[J].Journal of Mathemetical Study,2000,33(3):229-243.

[7]WANGJunping,YE Xiu.Supperconvergenceof finite element app roximations fo r the Stokesp roblem by least-squares surface fitting[J].SIAM Journalon Numerical Analysis,2001,30(3):1001-1013.

[8]DOUGLASJ,DUPONT T.Superconvergence fo r Galerkin methods fo r the two-point boundary p roblem via local p rojections[J].M uner Math,1973,21:270-278.

[9]L IN Qun,L IN Jiafu.Finite elementmethods:accuracy and imp rovement[M].Beijing:Science Press,2006.

[10]林群,严宁宁.高校有限元构造与分析[M].保定:河北大学出版社,1996.

(责任编辑:王兰英)

Optim ized Result for the Stokes Problem

GAO Shao-qin
(College of Mathematics and Computer Science,Hebei University,Baoding 071002,China)

The objective is to study the superconvergence result for the least-squares finite element app roximation of Stokesp rop lem.An op timized result of velocity fo r the least squares finite element app roxiationsof Stokes p roblem using a least-squares surface fitting method is developed.In theo ry the accuracy of velocity w ill be developed w ith the increasing o rder of piece-w ise polynomialsof fitting space.The result is based on som e regularity assump tion fo r the Stokes p roblem.

Stokes p roblem;the least-squares finite element method;velocity;op timized

O 427.4

A

1000-1565(2010)06-0613-04

2009-09-15

河北省教育厅自然科学基金资助项目(2009107)

高少芹(1970-),女,河北衡水人,河北大学副教授,主要从事计算数学方面的研究.