Sobolev方程的全离散有限体积元格式及数值模拟

doi:10.12286/jssx.2012.2.163

计算数学 ›› 2012, Vol. 34 ›› Issue (2): 163-172.DOI: 10.12286/jssx.2012.2.163

Sobolev方程的全离散有限体积元格式及数值模拟

李宏¹, 罗振东², 安静³, 孙萍³

1. 内蒙古大学数学科学学院, 呼和浩特 010021;
2. 华北电力大学数理学院, 北京 102206;
3. 贵州师范大学数学与计算机科学学院, 贵阳 550001

收稿日期:2011-07-05 出版日期:2012-05-15 发布日期:2012-05-20
基金资助:
国家自然科学基金(批准号: 11061009和11061021)、河北省自然科学基金(批准号: A2010001663)、贵州省科技计划项目(批准号: 黔科合J字[2011]2367)和内蒙古自治区高等学校研究项目(批准号: NJ10006)资助.

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李宏, 罗振东, 安静, 孙萍. Sobolev方程的全离散有限体积元格式及数值模拟[J]. 计算数学, 2012, 34(2): 163-172.

Li Hong, Luo Zhendong, An Jing, Sun Ping. A FULLY DISCRETE FINITE VOLUME ELEMENT FORMULATION FOR SOBOLEV EQUATION AND NUMERICAL SIMULATIONS[J]. Mathematica Numerica Sinica, 2012, 34(2): 163-172.

A FULLY DISCRETE FINITE VOLUME ELEMENT FORMULATION FOR SOBOLEV EQUATION AND NUMERICAL SIMULATIONS

Li Hong¹, Luo Zhendong², An Jing³, Sun Ping³

1. School of Mathematical Sciences, Inner Mongolia University, Huhhot 010021, China;
2. School of Mathematics and Physics, North China Electric Power University, Beijing 102206, China;
3. School of Mathematics and Computer Science, Guizhou Normal University, Guiyang, 550001, China

Received:2011-07-05 Online:2012-05-15 Published:2012-05-20

本文研究二维Sobolev方程的有限体积元方法, 给出一种全离散化有限体积元格式及其有限体积元解的误差估计,并用数值例子说明数值计算的结果与理论结果是相吻合的, 进一步说明了有限体积元方法比其他数值方法更优越.

关键词： Sobolev方程, 有限体积元格式, 全离散格式, 误差估计

In this paper, a finite volume element method for 2D Sobolev equation is studied and a fully discrete finite volume element formulation and error estimates are derived. Some numerical examples illustrate the fact that the results of numerical computation are consistent with theoretical conclusions. Moreover, it is shown that the finite volume element method is more advantageous than others for finding numerical solutions of Sobolev equation.

Key words: Sobolev equation, finite volume element formulation, fully discrete formulation, error estimate

MR(2010)主题分类:

65N30
65M30

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[1] Barenblett G I, Zheltov I P, Kochian I N. Basic concepts in the theory of homogeneous liquids in fissured rocks[J]. J. Appl. Math. Mech., 1990, 24(5): 1286–1303.

[2] 施德明. 非线性湿气迁移方程的边值问题[J]. 应用数学学报, 1990, 13(1): 33--40.

[3] Ting T W. A cooling process according to two-temperature theory of heat conduction[J]. J. Math. Anal. Appl., 1974, 45(1): 23–31.

[4] Cai Z, McCormick S. On the accuracy of the finite volume element method for diffusion equations on composite grid[J]. SIAM Journal on Numerical Analysis, 1990, 27(3): 636–655.

[5] Süli E. Convergence of finite volume schemes for Poisson’s equation on nonuniform meshes[J]. SIAM Journal on Numerical Analysis, 1991, 28(5): 1419–1430.

[6] Jones W P, Menziest K R. Analysis of the cell-centred finite volume method for the diffusion equation[J]. Journal of Computational Physics, 2000, 165(1): 45–68.

[7] Bank R E, Rose D J. Some error estimates for the box methods[J]. SIAM Journal on Numerical Analysis, 1987, 24(4): 777–787.

[8] Li R H, Chen Z Y, Wu W. Generalized Difference Methods for Differential Equations-Numerical Analysis of Finite Volume Methods[M]. Monographs and Textbooks in Pure and Applied Mathematics 226. Marcel Dekker Inc.: New York, 2000.

[9] Blanc P, Eymerd R, Herbin R. A error estimate for finite volume methods for the Stokes equations on equilateral triangular meshes[J]. Numerical Methods for Partial Differential Equations, 2004, 20(6): 907–918.

[10] Li J, Chen Z X. A new stabilized finite volume method for the stationary Stokes equations[J]. Adv. Comput. Math., 2009, 30(2): 141–152.

[11] Shen L H, Li J, Chen Z X. Analysis of stabilized finite volume method for the transient stationary Stokes equations[J]. International Journal of Numerical Analysis and Modeling, 2009, 6(3): 505– 519.

[12] 曹艳华. 二维线性Sobolev方程广义差分方法[J]. 计算数学, 2005, 27(3): 243--256.

[13] 赵洁. 二维Sobolev方程的有限体积元法[C]. 内蒙古大学硕士论文, 2010.

[14] Adams R A. Sobolev Space[M]. New York: Academic Press, 1975.

[15] 罗振东. 混合有限元法基础及其应用[M]. 北京: 科学出版社, 2006.

[16] Brezzi F, Fortin M. Mixed and Hybrid Finite Element Methods[M]. New York: Springer–Verlag, 1991.

[17] Ciarlet P G. The Finite Element Method for Elliptic Problems[M]. New York: North–Holland, Amsterdam, 1978.

[18] Temam R. Navier–Stokes equations (3rd)[M]. New York: North–Holland, Amsterdam, 1984.

[19] Luo Z D, Li H, An J, Sun P. A reduced finite volume element formulation based on POD method and numerical simulation for two-dimensional Sobolev equation[J]. Arch. Comput. Method. Eng., 2011 (in press).

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摘要

Sobolev方程的全离散有限体积元格式及数值模拟

A FULLY DISCRETE FINITE VOLUME ELEMENT FORMULATION FOR SOBOLEV EQUATION AND NUMERICAL SIMULATIONS

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