• 论文 •

### 一阶双曲问题间断有限元的后验误差分析

1. 东北大学数学系, 沈阳 110004
• 收稿日期:2011-12-07 出版日期:2012-05-15 发布日期:2012-05-20
• 基金资助:

国家自然科学基金资助项目(No.11071033).

Zhang Tie, Li Zheng. A POSTERIORI ERROR ANALYSIS OF DISCONTINUOUS GALERKIN METHOD FOR FIRST ORDER HYPERBOLIC PROBLEMS[J]. Mathematica Numerica Sinica, 2012, 34(2): 215-224.

### A POSTERIORI ERROR ANALYSIS OF DISCONTINUOUS GALERKIN METHOD FOR FIRST ORDER HYPERBOLIC PROBLEMS

Zhang Tie, Li Zheng

1. Department of Mathematics, Northeastern University, Shenyang 110004, China
• Received:2011-12-07 Online:2012-05-15 Published:2012-05-20

Until now, the a posteriori error estimates of finite element methods for first order hyperbolic problems are yet far from resolving well. In this paper, we propose a new a posteriori error analysis method for the discontinuous finite element approximates to first order hyperbolic problems in d space dimensions by using polynomials of order k, and then we establish efficiency residual-based a posteriori error estimates in the DG norm which is stronger then the L2 norm. The validity of our theoretical analysis is verified by some numerical experiments. Our method is also available for finite element approximations to other variational problems.

MR(2010)主题分类:

()
 [1] Mekchay K, Nochetto R H. Convergence of adaptive finite element methods for general second order linear elliptic PDEs[J]. SIAM J Numer Anal, 2005, 43: 1803-1827.[2] Yan N N. Superconvergence Analysis and A Posteriori Error Estimation in Finite Element Methods[ M]. Beijing: Science Press, 2008.[3] Houston P, Rannacher R, Süli E. A posteriori error analysis for Stabilized finite element approximations of transport problems[J]. Comput Methods Appl Mech Engrg, 2000, 190: 1483-1508.[4] Houston P, Süli E. hp-adaptive discontinuous Galerkin finite element methods for first-order hyperbolic problems[J]. SIAM J Sci Comput, 2001, 23: 1226-1252.[5] Burnman E. A posteriori error estimation for interior penalty finite element approximations of the advection-reaction equation[J]. SIAM J Numer Anal 2009, 47: 3584-3670.[6] Sangalli G. Robust a posteriori estimators for advection-diffusion-reaction problems[J]. Math Comp, 2008, 77: 41-70.[7] Braess D, Verfürth R. A posteriori error estimators for the Raviart-Thomas element[J]. SIAM J Numer Anal, 1996, 33: 2431-2444.[8] Friedrichs K. Symmetric positive linear differential equations[J]. Comm Pure Appl Math, 1958, 11: 333-418.[9] Reed W H, Hill T R. Triangular mesh methods for neutron transport equation[R]. Tech. Report LA-Ur-73-479, Los Alamos Scientific Laboratory, 1973.[10] Lesaint P, Raviart R A. On a finite element method for solving the neutron transport equation. In: Mathematical Aspects of Finite Elements in Partial Differential Equations[M]. C. de Boor, ed, New York: Academic Press, 1974, 89-145.[11] Johnson C, Pitkaranta J. An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation[J]. Math Comp, 1986, 46: 1-26.[12] Guzman J. Local analysis of discontinuous Galerkin methods applied to singularty perturbed problems[J]. J. Numer. Math., 2006, 14: 41-56.[13] Peterson T E. A note on the convergence of the discontinuous Galerkin method for a scalar hyperbolic equation[J]. SIAM J Numer Anal, 1991, 28: 133-140.[14] 严宁宁, 周爱辉, 林群, 基于特征相关网格的不连续Galerkin法[J], 应用数学与计算数学学报, 1992, 6: 76-83.[15] 林群, 严宁宁, 高效有限元构造与分析[M], 河北大学出版社, 河北, 1996, 119-120.[16] Cockburn B, Dong B, Guzm′an J. Optimal convergence of the original DG method for the transport-reaction equation on special meshes[J]. SIAM J Numer Anal, 2008, 46: 1250-1265.[17] Cockburn B, Dong B, Guzm′an J, Qian J L. Optimal convergence of the original DG method special meshes for variable transport velocity[J]. SIAM J Numer Anal, 2010, 48: 139-146.
 [1] 祝鹏, 尹云辉, 杨宇博. 奇异摄动问题内罚间断有限方法的最优阶一致收敛性分析[J]. 计算数学, 2013, 35(3): 323-336. [2] 张铁, 冯男, 史大涛. 求解椭圆边值问题惩罚形式的间断有限元方法[J]. 计算数学, 2010, 32(3): 275-284. [3] 孙澈,汤怀民,吴克俭. 一阶双曲问题的间断流线扩散法[J]. 计算数学, 1998, 20(1): 35-44.