一类非线性对流扩散方程两重网格特征有限元方法及误差估计

数学物理学报 ›› 2014, Vol. 34 ›› Issue (3): 643-654.

一类非线性对流扩散方程两重网格特征有限元方法及误差估计

陈传军|赵鑫

烟台大学数学与信息科学学院山东烟台 264005

收稿日期:2012-06-19 修回日期:2013-11-12 出版日期:2014-06-25 发布日期:2014-06-25
基金资助:
国家自然科学基金$(11301456)、山东省自然科学基金(ZR2010AQ010, ZR2011AQ021)和山东省高等学校青年骨干教师国内访问学者项目经费资助

Analysis of a Two-grid Method for Characteristic Finite Element Solutions of Nonlinear Convection-diffusion Equations

CHEN Chuan-Jun, ZHAO Xin

School of Mathematics and Informational Science, Yantai University, Shandong Yantai 264005

Received:2012-06-19 Revised:2013-11-12 Online:2014-06-25 Published:2014-06-25
Supported by:
国家自然科学基金$(11301456)、山东省自然科学基金(ZR2010AQ010, ZR2011AQ021)和山东省高等学校青年骨干教师国内访问学者项目经费资助

摘要/Abstract

摘要：

主要研究了一类非线性对流扩散方程的全离散特征有限元方法的两重网格算法及其误差估计. 首先在网格步长为$H$的粗网格上计算一个较小的非线性问题, 然后利用一阶牛顿迭代和粗网格解将网格步长为h的细网格上的非线性问题转化为线性问题求解. 由于非线性问题的求解仅在粗网格上进行, 该两重网格算法可以节省大量的计算工作量, 同时具有较高的
精度, 证明了该两重网格算法L²模先验误差估计结果为O(Δt +h²+H^4-d/2), 其中d为空间维数.

关键词: 两重网格, 有限元方法, 非线性, 误差估计

Abstract:

A two-grid characteristic finite element approximation is presented and discussed for nonlinear convection-diffusion equations. Piecewise linear trial functions are used to develop the finite element scheme. In this two-grid scheme, the full nonlinear problem is solved only on a coarse grid with grid size H. The nonlinearities are expanded by one Newton-like iteration about the coarse grid solution on a fine gird of size h. The resulting linear system is solved only on the fine grid. A prior error estimates are derived with the L²-norm O(?t +h²+H^4-d/2), where d≥1 is the spacial dimension.

Key words: Two-grid, Finite element method, Nonlinear, Error estimates

中图分类号:

65N12

陈传军, 赵鑫. 一类非线性对流扩散方程两重网格特征有限元方法及误差估计[J]. 数学物理学报, 2014, 34(3): 643-654.

CHEN Chuan-Jun, ZHAO Xin. Analysis of a Two-grid Method for Characteristic Finite Element Solutions of Nonlinear Convection-diffusion Equations[J]. Acta mathematica scientia,Series A, 2014, 34(3): 643-654.

参考文献

[1] Chen Z. Characteristic-nonconforming finite-element methods for advection-dominated diffusion problems. Comput Math Appl, 2004, 48: 1087--1100

[2]} Douglas J Jr, Huang C S, Pereira F. The modified method of characteristics with adjusted advection. Numer Math, 1999, 83: 353--369

[3] Wang H. An optimal-order error estimate for MMOC and MMOCAA schemes for multidimensional advection-reaction equations. Numer Meth PDEs, 2002, 18: 69--84

[4] Douglas J Jr, Russell T F. Numerical methods for convection-dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures. SIAM J Numer Anal, 1982, 19: 871--885

[5] Russell T F. Time stepping along characteristics with incomplete iteration for a Galerkin approximation of miscible displacement in porous media. SIAM J Numer Anal, 1985, 22: 970--1013

[6] Xu J. A novel two-grid method for semilinear elliptic equations. SIAM J Sci Comput, 1994, 15: 231--237

[7] Xu J. Two-grid discretization techniques for linear and nonlinear PDEs. SIAM J Numer Anal, 1996, 33: 1759--1777

[8] Dawson C N, Wheeler M F. Two-grid methods for mixed finite element approximations of nonlinear parabolic
equations. Contemp Math, 1994, 180: 191--203

[9] Dawson C N, Wheeler M F, Woodward C S. A two-grid finite difference scheme for nonlinear parabolic equations.
SIAM J Numer Anal, 1998, 35: 435--452

[10] Wu L, Allen M B. A two-grid method for mixed finite-element solutions of reaction-diffusion equations. Numer Meth PDEs, 1999, 15: 589--604

[11] Chen Y, Huang Y, Yu D. A two-grid method for expanded mixed finite-element solution of semilinear reaction-diffusion equations. Int J Numer Meth Engng, 2003, 57: 139--209

[12] Chen Y, Liu H, Liu S. Analysis of two-grid methods for reaction diffusion equations by expanded mixed finite element methods. Int J Numer Meth Engng, 2007, 69: 408--422

[13] Chen L, Chen Y. Two-grid method for nonlinear reaction-diffusion equations by mixed finite element methods. J Sci Comput, 2011, 49: 383--401

[14] Bi C, Ginting V. Two-grid finite volume element method for linear and nonlinear elliptic problems. Numer Math,
2007, 108: 177--198

[15] Chen C, Yang M, Bi C. Two-grid methods for finite volume element approximations of nonlinear parabolic equations. J Comput Appl Math, 2009, 228: 123--132

[16] Chen C, Liu W. Two-grid finite volume element methods for semilinear parabolic problems. Appl Numer Math,
2010, 60: 10--18

[17] Chen C, Liu W. A two-grid method for finite volume element approximations of second-order nonlinear hyperbolic equations. J Comput Appl Math, 2010, 233: 2975--2984

[18] Qin X, Ma Y. Two-grid method for characteristics finite-element solution of 2d nonlinear convection-dominated diffusion problem. Appl Math Comput, 2005, 165: 419--431

[19] Chen C, Bi C. Two-grid methods for characteristic finite volume element solution of semilinear convection-diffusion equations. Appl Math Comput, 2010, 217: 1896--1906

[20] 袁益让, 王宏. 非线性双曲型方程有限元方法的误差估计. 系统科学与数学,1985, 5: 161--171

[21] Thom\'{e}e V. Galerkin Finite Element Methods for Parabolic Problems. Berlin: Springer-verlag, 2006

[22] Hayes L J. A modified backward time discretization for nonlinear parabolic equations using patch approximation.
SIAM J Numer Anal, 1981, 18: 781--793

[23] 李潜. 非线性抛物方型程的全离散有限元方法. 应用数学与计算数学学报, 1988, 2: 50--58

[24] Yang D. Least-squares mixed finite element methods for nonlinear parabolic problems. J Comput Math, 2002, 20: 153--164