• 论文 • 上一篇    下一篇

求解二维时谐Maxwell方程的一种混合有限元新格式

曹济伟1,2   

  1. 1. 河南财经政法大学数学与信息科学学院, 郑州 450000;
    2. 南开大学数学科学学院, 天津 300071
  • 收稿日期:2016-01-22 出版日期:2016-12-15 发布日期:2016-10-13
  • 基金资助:

    国家自然科学基金11071132,11171168,91430106,11571266资助.

曹济伟. 求解二维时谐Maxwell方程的一种混合有限元新格式[J]. 计算数学, 2016, 38(4): 429-441.

Cao Jiwei. A NEW MIXED FINITE ELEMENT SCHEME FOR SOLVING 2D TIME-HARMONIC MAXWELL-TYPE PROBLEM[J]. Mathematica Numerica Sinica, 2016, 38(4): 429-441.

A NEW MIXED FINITE ELEMENT SCHEME FOR SOLVING 2D TIME-HARMONIC MAXWELL-TYPE PROBLEM

Cao Jiwei1,2   

  1. 1. School of Mathematics and Information Science, Henan University of Economics and Law, Zhengzhou 450000, China;
    2. School of Mathematical Sciences, Nankai University, Tianjin 300071, China
  • Received:2016-01-22 Online:2016-12-15 Published:2016-10-13
本文,我们提出一种新的求解二维时谐Maxwell方程的H1-协调节点连续混合有限元格式.由于加上若干稳定化项和投影项,得到的混合变分形式是稳定的.我们证明了双线性形式满足连续性,Kh-强制性和Inf-Sup条件,因此,解是存在唯一的.此外,我们也给出了拟优的误差估计和相应的收敛阶.
In this paper, we propose a new H1-conforming nodal-continuous mixed finite element scheme. Several stabilizations and projections are added into the mixed finite element scheme and thus the stability of the variational system is achieved. We prove that the bilinear forms satisfy continuity, Kh-coercivity and Inf-Sup condition, and hence the existence and uniqueness of solution hold. Furthermore, quasi-optimal error estimates and convergence rate are derived.

MR(2010)主题分类: 

()
[1] Adams R A and Fournier J. Sobolev Spaces[M]. Second Edition, Academic Press, New York, 2003.

[2] Monk P. Finite Element Methods for Maxwell Equations[M]. Clarendon Press, Oxford, 2003.

[3] Buffa A. Remarks on the discretization of some noncoercive operator with applications to heterogeneous maxwell equations[J]. SIAM J. Numer. Anal., 2005, 43(1):1-18.

[4] Nédélec J. Mixed finite elements in R3[J]. Numer. Math., 1980, 35(3):315-341.

[5] Arnold D N, Brezzi F and Marini L D. Unified analysis of discontinuous Galerkin methods for elliptic problems[J]. SIAM J. Numer. Anal., 2001, 39(5):1749-1779.

[6] Riviére B. Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations:Theory and Implementation[M]. SIAM, 2008.

[7] Duan H Y, Li S. Tan Roger C.E. and Zheng W.Y., A delta-regularization finite element method for a double curl problem with divergence-free constraint[J]. SIAM J. Numer. Anal., 2012, 50(6):3208-3230.

[8] Duan H Y, Lin P and Tan Roger C E. C0 elements for generalized indefinite Maxwell's equations[J]. Numer. Math., 2012, 122(1):61-99.

[9] Houston P, Perugia I, Schneebeli A and Schötzau D. Interior penalty method for indefinite timeharmonic Maxwell equations[J]. Numer. Math., 2005100(3):485-518.

[10] Grote M J, Schneebeli A and Schötzau D. Interior penalty discontinuous Galerkin method for Maxwell's equations:Energy norm error estimates[J]. J. Comp. Appl. Math., 2007, 204(2):375-386.

[11] Cockburn B, Li F and Shu C W. Locally divergence-free discontinuous Galerkin methods for the Maxwell equations[J]. J. Comput. Phys., 2004, 194(2):588-610.

[12] Otin R. Regularized Maxwell equations and nodal finite elements for electromagnetic field computations[J]. Electromagnetics, 2010, 30(1):190-204.

[13] Costabel M and Dauge M. Weighted regularization of Maxwell equations in polyhedral domains[J]. Numer. Math., 2002, 93(2):239-277.

[14] Brezzi F and Fortin M. Mixed and Hybrid Finite Element Methods[M]. Springer-Verlag, NewYork, 1991.

[15] Girault V and Raviart P A. Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms[M]. Springer-Verlag, Berlin, 1986.

[16] Duan H Y, Lin P and Tan Roger C E. Error estimates for a vectorial second-order elliptic eigenproblem by the local L2 projected C0 finite element method[J]. SIAM J. Numer. Anal., 2013, 51(3):1678-1714.

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

[18] Duan H Y, Jia F, Lin P and Tan Roger C E. The local L2 projected C0 finite element method for Maxwell problem[J]. SIAM J. Numer. Anal., 2009, 47(2):1274-1303.

[19] Brenner S C and Scott L R. The Mathematical Theory of Finite Element Methods[M]. Third Edition, Springer-Verlag, New-York, 2008.

[20] Clément P. Approximation by finite element functions using local regularization[J]. RAIRO Anal. Numer., 1975, 2(R-2):77-84.

[21] Xue Y, Duan H Y and Zhang Q. A new and simple implementation of the element-local L2-projected continuous finite element method[J]. Appl. Math. Comput., 2014, 228:170-183.

[22] Duan H Y, Tan Roger C E, Yang S Y and You C S. Computation of Maxwell singular solution by nodal-continuous elements[J]. J. Comput. Phys., 2014, 268:63-83.

[23] Buffa A and Ciarlet P Jr. On traces for functional spaces related to Maxwell's equations Part Ⅱ:Hodge decompositions on the boundary of Lipschitz polyhedra and applications[J]. Math. Meth. Appl. Sci., 2001, 24(1):31-48.

[24] Dhia A B, Bonnet-Ben, Hazard C and Lohrengel S. A Singular Field Method For the Solution of Maxwell's Equations in Polyhedral Domains[J]. SIAM J. Appl. Math., 1999, 59(6):2028-2044.

[25] Duan H Y, Lin P, Saikrishnan P and Tan Roger C E. A least-squares finite element method for the magnetostatic problem in a multiply connected Lipschitz domain[J]. SIAM J. Numer. Anal., 2007, 45(6):2537-2563.

[26] Duan H Y, Lin P and Tan Roger C E. Analysis of a continuous finite element method for H(curl, div)-elliptic interface problem[J]. Numer. Math., 2013, 123(4):671-707.

[27] Boffi D, Brezzi F and Fortin M. Mixed Finite Element Methods and Applications[M]. Springer, 2013.

[28] Arnold D N, Brezzi F, Cockburn B and Marini D. Discontinuous Galerkin Methods for Elliptic Problems[M]. Lect. Notes Comput. Sci. Eng., Springer, Berlin, 2000, 11:89-101.
[1] 洪庆国, 刘春梅, 许进超. 一种抽象的稳定化方法及在非线性不可压缩弹性问题上的应用[J]. 计算数学, 2020, 42(3): 298-309.
[2] 张然. 弱有限元方法在线弹性问题中的应用[J]. 计算数学, 2020, 42(1): 1-17.
[3] 王俊俊, 李庆富, 石东洋. 非线性抛物方程混合有限元方法的高精度分析[J]. 计算数学, 2019, 41(2): 191-211.
[4] 卢培培, 许学军. 高波数波动问题的多水平方法[J]. 计算数学, 2018, 40(2): 119-134.
[5] 石东洋, 史艳华, 王芬玲. 四阶抛物方程H1-Galerkin混合有限元方法的超逼近及最优误差估计[J]. 计算数学, 2014, 36(4): 363-380.
[6] 周琴, 潘雪琴, 冯民富. 对流占优的Sobolev方程的投影稳定化有限元方法[J]. 计算数学, 2014, 36(1): 99-112.
[7] 李先崇, 孙萍, 安静, 罗振东. 粘弹性方程一种新的分裂正定混合元法[J]. 计算数学, 2013, 35(1): 49-58.
[8] 石东洋, 唐启立, 董晓靖. 强阻尼波动方程的H1-Galerkin混合有限元超收敛分析[J]. 计算数学, 2012, 34(3): 317-328.
[9] 司红颖, 陈绍春. 双调和方程混合元的一种新格式[J]. 计算数学, 2012, 34(2): 173-182.
[10] 罗兴钧, 李繁春, 杨素华. 最优投影策略下解病态积分方程的快速迭代算法[J]. 计算数学, 2011, 33(1): 1-14.
[11] 马昌凤,梁国平,刘韶鹏. 一类波动方程有限元投影格式的误差分析[J]. 计算数学, 2002, 24(4): 501-512.
[12] 黄兰洁. 关于非定常不可压Navier-Stokes方程的时间高精度隐式差分方法[J]. 计算数学, 2002, 24(2): 197-218.
阅读次数
全文


摘要