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高波数Helmholtz方程的有限元方法和连续内罚有限元方法

本站小编 Free考研考试/2021-12-27

武海军
南京大学数学系, 南京 210093
收稿日期:2017-08-31出版日期:2018-06-15发布日期:2018-05-15


基金资助:国家自然科学基金(11525103,91630309,11621101)资助.


FEM AND CIP-FEM FOR HELMHOLTZ EQUATION WITH HIGH WAVE NUMBER

Wu Haijun
Department of Mathematics, Nanjing University, Nanjing 210093, China
Received:2017-08-31Online:2018-06-15Published:2018-05-15







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本文介绍高波数Helmholtz方程的有限元方法和连续内罚有限元方法.将以线性元情形为例,给出方法的明显依赖于波数k的预渐近稳定性和误差分析.我们将介绍三种证明方法.我们还讨论了内罚有限元方法的罚参数的选取以显著减少方法的污染误差.最后还给出数值例子验证理论结果.
MR(2010)主题分类:
65N12
65N15
65N30
78A40

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[1] Ainsworth M. Discrete dispersion relation for hp-version finite element approximation at high wave number[J]. SIAM J. Numer. Anal., 2004, 42(2):553-575.

[2] Aziz A and Kellogg R. A scattering problem for the Helmholtz equation. In Advances in Computer Methods for Partial Differential Equations-Ⅲ, 1979, 93-95.

[3] Babuška I and Sauter S. Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers?[J]. SIAM Rev., 2000, 42(3):451-484.

[4] Brenner S and Scott R. The Mathematical Theory of Finite Element Methods, volume 15. Springer Science & Business Media, 2007.

[5] Burman E, Zhu L and Wu H. Linear continuous interior penalty finite element method for Helmholtz equation with high wave number:One-dimensional analysis[J]. Numer. Meth. Par. Diff. Equ., 2016, 32:1378-1410.

[6] Chen H, Lu P and Xu X. A hybridizable discontinuous Galerkin method for the Helmholtz equation with high wave number[J]. SIAM J. Numer. Anal., 2013, 51:2166-2188.

[7] Chen H, Wu H and Xu X. Multilevel preconditioner with stable coarse grid corrections for the helmholtz equation[J]. SIAM J. Sci. Comput., 2015, 37:A221-A244.

[8] Chen Z, Wu T and Yang H. An optimal 25-point finite difference scheme for the Helmholtz equation with PML[J]. J. Comput. Appl. Math., 2011, 236:1240-1258.

[9] Chen Z and Xiang X. A source transfer domain decomposition method for Helmholtz equations in unbounded domain[J]. SIAM J. Numer. Anal., 2013, 51(4):2331-2356.

[10] Demkowicz L, Gopalakrishnan J, Muga I and Zitelli J. Wavenumber explicit analysis of a DPG method for the multidimensional Helmholtz equation[J]. Comput. Methods Appl. Mech. Engrg., 2012, 214(12):126-138.

[11] Deraemaeker A, Babuška I and Bouillard P. Dispersion and pollution of the FEM solution for the Helmholtz equation in one, two and three dimensions[J]. Internat. J. Numer. Methods Engrg., 1999, 46:471-499.

[12] Douglas J and Dupont T. Interior Penalty Procedures for Elliptic and Parabolic Galerkin Methods. Lecture Notes in Physics, 1976, 58:207-216.

[13] Du Y and Wu H. An improved pure source transfer domain decomposition method for Helmholtz equations in unbounded domain. ArXiv e-prints, 2015.

[14] Du Y and Wu H. Preasymptotic error analysis of higher order FEM and CIP-FEM for Helmholtz equation with high wave number[J]. SIAM J. Numer. Anal., 2015, 53(2):782-804.

[15] Du Y and Zhu L. Preasymptotic error analysis of high order interior penalty discontinuous Galerkin methods for the Helmholtz equation with high wave number[J]. J. Sci. Comput., 2016, 67:130-152.

[16] Engquist B and Runborg O. Computational high frequency wave propagation[J]. Acta numerica, 2003, 12:181-266.

[17] Engquist B and Ying L. Sweeping preconditioner for the Helmholtz equation:hierarchical matrix representation[J]. Comm. Pure Appl. Math., 2011, 64(5):697-735.

[18] Engquist B and Ying L. Sweeping preconditioner for the Helmholtz equation:moving perfectly matched layers[J]. Multiscale Model. Simul., 2011, 9:686-710.

[19] Feng X and Wu H. Discontinuous Galerkin methods for the Helmholtz equation with large wave numbers[J]. SIAM J. Numer. Anal., 2009, 47(4):2872-2896.

[20] Feng X and Wu H. hp-discontinuous Galerkin methods for the Helmholtz equation with large wave number[J]. Math. Comp., 2011, 80(276):1997-2024.

[21] Feng X and Wu H. An absolutely stable discontinuous Galerkin method for the indefinite timeharmonic Maxwell equations with large wave number[J]. SIAM J. Numer. Anal., 2014, 52:2356-2380.

[22] Hiptmair R, Moiola A and Perugia I. Plane wave discontinuous Galerkin methods for the 2d Helmholtz equation:Analysis of the p-version[J]. SIAM J. Numer. Anal., 2011, 49:264-284.

[23] Hiptmair R, Moiola A and Perugia I. A Survey of Trefftz Methods for the Helmholtz Equation, pages 237-279. Springer International Publishing, Cham, 2016.

[24] Hu Q and Zhang H. Substructuring preconditioners for the systems arising from plane wave discretization of Helmholtz equations[J]. SIAM J. Sci. Comput., 2016, 38:A2232-A2261.

[25] Ihlenburg F. Finite element analysis of acoustic scattering, volume 132 of Applied Mathematical Sciences. Springer-Verlag, New York, 1998.

[26] Ihlenburg F and Babuska I. Finite element solution of the Helmholtz equation with high wave number part Ⅱ:the hp version of the FEM[J]. SIAM J. Numer. Anal., 1997, 34(1):315-358.

[27] Ihlenburg F and Babuška I. Finite element solution of the Helmholtz equation with high wave number. I. The h-version of the FEM[J]. Comput. Math. Appl., 1995, 30(9):9-37.

[28] Douglas J, Santos J and Sheen D. Approximation of scalar waves in the space-frequency domain[J]. Math. Models Methods Appl. Sci., 1994, 4:509-531.

[29] Li Y and Wu H. Fem and cip-fem for helmholtz equation with high wave number and pml truncation. Submitted, 2017.

[30] Melenk J, Parsania A and Sauter S. General DG-methods for highly indefinite Helmholtz problems[J]. J. Sci. Comp., 2013, 57(3):536-581.

[31] Melenk J and Sauter S. Convergence analysis for finite element discretizations of the Helmholtz equation with Dirichlet-to-Neumann boundary conditions[J]. Math. Comp., 2010, 79(272):1871-1914.

[32] Melenk J and Sauter S. Wavenumber explicit convergence analysis for Galerkin discretizations of the Helmholtz equation[J]. SIAM J. Numer. Anal., 2011, 49(3):1210-1243.

[33] Moiola A and Spence E. Is the Helmholtz equation really sign-indefinite?[J]. SIAM Rev., 2014, 56:274-312.

[34] Schatz A. An observation concerning Ritz-Galerkin methods with indefinite bilinear forms[J]. Math. Comp., 1974, 28:959-962.

[35] Shen J and Wang L. Analysis of a spectral-Galerkin approximation to the Helmholtz equation in exterior domains[J]. SIAM J. Numer. Anal., 2007, 45(5):1954-1978.

[36] Thompson L. A review of finite-element methods for time-harmonic acoustics[J]. J. Acoust. Soc. Amer., 2006, 119:1315-1330.

[37] Wang K and Wong Y. Is pollution effect of finite difference schemes avoidable for multi-dimensional Helmholtz equations with high wave numbers?[J]. Commun. Comput. Phys., 2017, 21:490-514.

[38] Wu H. 高波数helmholtz方程的内罚有限元方法[J].中国科学:数学, 2012, 42:429-444.

[39] Wu H. Pre-asymptotic error analysis of CIP-FEM and FEM for the Helmholtz equation with high wave number. Part I:linear version[J]. IMA J. Numer. Anal., 2013, 34(3):1266-1288.

[40] Zhu L and Wu H. Preasymptotic error analysis of CIP-FEM and FEM for Helmholtz equation with high wave number. part Ⅱ:hp version[J]. SIAM J. Numer. Anal., 2013, 51(3):1828-1852.

[41] Zienkiewicz O. Achievements and some unsolved problems of the finite element method[J]. Int. J. Numer. Meth. Engng., 2000, 47:9-28.

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