高少芹

(河北大學數(shù)學與計算機學院,河北保定 071002)

研究報告

STO KES問題的優(yōu)化結果

高少芹

(河北大學數(shù)學與計算機學院,河北保定 071002)

為了研究Stokes問題的最小二乘有限元近似解的超收斂結果,利用最小二乘曲面擬合的方法給出了Stokes問題的最小二乘有限元解中速度的優(yōu)化結果.該結果表明:隨著構成擬合空間的分片多項式次數(shù)的增加,理論上得到的速度的近似解精度越高.該結果是以Stokes問題的正則性為前提的.

Stokes問題;最小二乘有限元;速度;優(yōu)化

1 預備知識

近年來,最小二乘有限元方法越來越被關注,可以用于求解數(shù)學、物理問題[1-4].而有限元解的超收斂也是一個有趣且在工業(yè)問題的科學計算中非常有用的問題[5-10],但有關最小二乘有限元解的超收斂方面的文章較罕見.文獻[1]給出了Stokes問題的最小二乘有限元近似解中速度u的近似值uh的L2-范數(shù)誤差估計,本文將給出速度u的一種優(yōu)化的近似解,這種結果是利用文獻[6-7]中對于標準Galerkin方法提出并作出分析的最小二乘曲面擬合方法得到的.

考慮Stokes問題

2 主要結果

3 結論

[1]DUAN Huoyuan.On the velocity-p ressure-vo rticity least-squresm ixed finite elementmethod fo r the 3D-Stokes equations[J].SIAM J Numer Anal,2003,41:2114-2130.

[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,嚴寧寧.高校有限元構造與分析[M].保定:河北大學出版社,1996.

(責任編輯:王蘭英)

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

河北省教育廳自然科學基金資助項目(2009107)

高少芹(1970-),女,河北衡水人,河北大學副教授,主要從事計算數(shù)學方面的研究.