H(curl)-椭圆问题不连续Galerkin法的后验误差估计

邢小青; 钟柳强

doi:10.6054/j.jscnun.2012.06.003

H(curl)-椭圆问题不连续Galerkin法的后验误差估计

邢小青¹,
钟柳强^2, ,

华南师范大学数学科学学院

基金项目:

国家自然科学基金资助项目

详细信息

通讯作者:
钟柳强

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出版历程
- 收稿日期: 2012-03-24
- 修回日期: 2012-05-07
- 刊出日期: 2012-08-24

A POSTERIORI ERROR ESTIMATE OF DISCONTINUOUS GALERKIN METHODS FOR H(curl)-ELLIPTIC PROBLEMS

Xiao-qing Xing¹,
Liu-qiang ZHONG^2, ,

More Information

Corresponding author:
Liu-qiang ZHONG

摘要

摘要: 针对 Lipschitz 多面体区域上 -椭圆问题的不连续 Galerkin 法, 提出了一种新的基于残量型的后验误差估计, 并证明了该后验误差的一个上界估计. 其中问题的最困难性在于如何处理跳跃项中出现的局部网格尺寸的负次幂.
- 不连续Galerkin法 /
- 后验误差估计 /
- H(curl)-椭圆问题
Abstract: A new posteriori error estimate based on residual for discontinuous Galerkin discretizations of $\boldsymbol{H}(\boldsymbol{curl})$ -elliptic problems on Lipschitz polyhedron is proposed. The corresponding upper bound is proved, where one of the most difficult problem is how to deal with the presence of the negative power of the local mesh size in the jump term.
- discontinuous Galerkin methods /
- a posteriori error estimate /
- H(curl)-elliptic problems

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参考文献(1)

[1] BOSSAVIT A. Computational Electromagnetism. Variational Formulation, Complementarity, Edge Elments[M].Academic Press, San Diego, CA, 1998.
[2] CARSTENSEN C, HOPPE R. Unified framework for an a posteriori error analysis of non-standard fi- nite element approximations of H(curl)-elliptic problems[C]//IEEE International Conference on Elec- tromagnetics in Advanced Applications(ICEAA’09), Torino, Italy, 2009: 754-755.
[3] CARSTENSEN C, HOPPE R, SHARMA N, WARBURTON T. Adaptive hybridized interior penalty discontinuous Galerkin methods for H(curl)-elliptic problems [J]. Numer Math Theor Meth Appl, 2011 4(1):13-37
[4] HIPTMAIR R. Multigrid method for Maxwell’s equations [J]. SIAM J Numer Anal, 1999, 36(1):204- 225.
[5] HOUSTON P, PERUGIA I,SCHENEEBELI A, SCHOTZAU D. Interior penalty method for the indefinite time-harmonic Maxwell equations [J]. Numer Math, 2005, 100(3):485-518.
[6] HOUSTON P, PERUGIA I, SCHOTZAU D.Mixed discontinuous Galerkin approximation of the Maxwell operator [J]. SIAM J Numer Anal, 2005, 42(1):434-459.
[7] HOUSTON P, PERUGIA I, SCHOTZAU D. An a posteriori error indicator for discontinuous Galerkin discretizations of H(curl)-elliptic partial differential equations [J]. IMA J Numer Anal, 2007, 27(1):122-150.
[8] PERUGIA I, SCHOTZAU D, MONK P. Stabilized interior penalty methods for the time-harmonic Maxwell equations [J]. Comput Methods Appl Mech Eng, 2002,191(41-42):4675-4697.
[9] SCHOBERL J. A posteriori error estimates for Maxwell equations [J]. Math Comp, 2008, 77(262):633- 649.

[1] BOSSAVIT A. Computational Electromagnetism. Variational Formulation, Complementarity, Edge Elments[M].Academic Press, San Diego, CA, 1998. [2] CARSTENSEN C, HOPPE R. Unified framework for an a posteriori error analysis of non-standard fi- nite element approximations of H(curl)-elliptic problems[C]//IEEE International Conference on Elec- tromagnetics in Advanced Applications(ICEAA’09), Torino, Italy, 2009: 754-755. [3] CARSTENSEN C, HOPPE R, SHARMA N, WARBURTON T. Adaptive hybridized interior penalty discontinuous Galerkin methods for H(curl)-elliptic problems [J]. Numer Math Theor Meth Appl, 2011 4(1):13-37 [4] HIPTMAIR R. Multigrid method for Maxwell’s equations [J]. SIAM J Numer Anal, 1999, 36(1):204- 225. [5] HOUSTON P, PERUGIA I,SCHENEEBELI A, SCHOTZAU D. Interior penalty method for the indefinite time-harmonic Maxwell equations [J]. Numer Math, 2005, 100(3):485-518. [6] HOUSTON P, PERUGIA I, SCHOTZAU D.Mixed discontinuous Galerkin approximation of the Maxwell operator [J]. SIAM J Numer Anal, 2005, 42(1):434-459. [7] HOUSTON P, PERUGIA I, SCHOTZAU D. An a posteriori error indicator for discontinuous Galerkin discretizations of H(curl)-elliptic partial differential equations [J]. IMA J Numer Anal, 2007, 27(1):122-150. [8] PERUGIA I, SCHOTZAU D, MONK P. Stabilized interior penalty methods for the time-harmonic Maxwell equations [J]. Comput Methods Appl Mech Eng, 2002,191(41-42):4675-4697. [9] SCHOBERL J. A posteriori error estimates for Maxwell equations [J]. Math Comp, 2008, 77(262):633- 649.

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