| [1] Ihlenburg F. Finite Element Analysis of Acoustic Scattering[M]. Spring, NewYork, 1998.[2] Bao G, Sun W. A fast algorithm for the electromaginetic scattering from a large cavity[J]. SIAM J. Sci. Comput., 2005, 27:553-574.[3] Bao G, Yun K, Zhou Z. Stability of the scattering from a large electromagnetic cavity in two dimensions[J]. SIAM J. Math. Anal., 2012, 44:383-404.[4] Chen Z, Liu X. An adaptive perfectly matched layer technique for time-harmonic scattering problems[J]. SIAM J. Numer. Anal., 2005, 43(2):645-671.[5] Chen Z, Liang C, Xiang X. An anisotropic perfectly matched layer method for Helmholtz scattering problems with discontinuous wave number[J]. Inverse Probl. Imag., 2013, 7:663-678.[6] Ma J, Zhu J, Li M. The Galerkin boundary element method for exterior problems of 2-D Helmholtz equation with arbitrary wavenumber[J]. Engrg. Anal. Bound. Elem., 2010, 34:1058-1063.[7] Hsiao G, Liu F, Sun J, Xu L. A coupled BEM and FEM for the interior transmission problem in acoustics[J]. J. Comput. Appl. Math., 2011, 235(17):5213-5221.[8] Hsiao G, Xu L. A system of boundary integral equations for the transmission problem in acoustics[J]. Appl. Numer. Math., 2011, 61(9):1017-1029.[9] Wang Y, Ma F, Zheng E. Galerkin method for the scattering problem of a slit[J]. J. Sci. Comput., 2017, 70(1):192-209.[10] Wu H, Liu Y, Jiang W. Analytical integration of the moments in the diagonal form fast multipole boundary element method for 3-D acoustic wave problems[J]. Eng. Anal. Bound. Elem., 2012, 63(2):248-254.[11] Geng H, Yin T, Xu L. A priori error estimates of the DtN-FEM for the transmission problem in acoustics[J]. J. Comput. Appl. Math., 2017, 313:1-17.[12] Sun W, Ma F. An error estimate of the coupled finite-infinite element method for scattering from an arc[J]. Int. J. Numer. Anal. Model., 2014, 11(4):841-853.[13] Shen J, Wang L. Analysis of a spectral-Galerkin approximation to the Helmholtz equation in exterior domains[J]. SIAM J. Numer. Anal., 2007, 45:1954-1978.[14] Shen J, Wang L. Spectral approximation of the Helmholtz equation with high wave numbers[J]. SIAM J. Numer. Anal., 2005, 43:623-644.[15] Baruch G, Fibich G, Tsynkov S, Turkel E. Fourth order schemes for time-harmonic wave equations with discontinuous coefficients[J]. Commun. Comput. Phys., 2009, 5(2-4):442-455.[16] Britt S, Tsynkov S, Turkel E. A compact fourth order scheme for the Helmholtz equation in polar coordinates[J]. J. Sci. Comput., 2010, 54:26-47.[17] Britt S, Tsynkov S, Turkel E. Numerical simulation of time-harmonic waves in inhomogeneous media using compact high order schemes[J]. Commun. Comput. Phys., 2011, 9(3):520-541.[18] Chen Z, Wu T, Yang H. An optimal 25-point finite difference scheme for the Helmholtz equation with PML[J]. J. Comput. Appl. Math., 2011, 236(6):1240-1258.[19] Chen Z, Cheng D, Wu T. A dispersion minimizing finite difference scheme and preconditioned solver for the 3D Helmholtz equation[J]. J. Comput. Phys., 2012, 231(24):8152-8175.[20] Chen Z, Cheng D, Feng W, Wu T. An optimal 9-point finite difference scheme for the Helmholtz equation with PML[J]. Int. J. Numer. Anal. Model., 2013, 10:389-410.[21] Cheng D, Liu Z, Wu T. A multigrid-based preconditioned solver for the Helmholtz equation with a discretization by 25-point difference scheme[J]. Math. Comput. Simul., 2015, 117:54-67.[22] Cheng D, Tan X, Zeng T. A dispersion minimizing finite difference scheme for the Helmholtz equation based on point-weighting[J]. Comput. Math. Appl., 2017, 73(11):2345-2359.[23] Fernandes D. A. Loula, Quasi optimal finite difference method for Helmholtz problem on unstructured grids[J]. Int. J. Numer. Meth. Engng., 2010, 82:1244-1281.[24] Feng X, Li Z, Qiao Z. High order compact finite difference schemes for the Helmholtz equation with discontinuous coefficients[J]. J. Comput. Math., 2011, 29:324-340.[25] Feng X. A high-order compact scheme for the one-dimensional Helmholtz equation with a discontinuous coefficient[J]. Int. J. Comput. Math., 2012, 89(5):618-624.[26] Fu Y. Compact fourth-order finite difference schemes for Helmholtz equation with high wave numbers[J]. J. Comput. Math., 2008, 26:98-111.[27] Guo R, Wang K, Xu L. Efficient finite difference methods for acoustic scattering from circular cylindrical obstacle[J]. Int. J. Numer. Anal. Model., 2016, 13:986-1002.[28] Harari I, Turkel E. Accurate finite difference methods for time-harmonic wave propagation[J]. J. Comput. Phys., 1995, 119:252-270.[29] Jo C, Shin C, Suh J. An optimal 9-point, finite-difference, frequency-space 2-D scalar wave extrapolator[J]. Geophysics, 1996, 61:529-537.[30] Lambe L, Luczak R, Nehrbass J. A new finite difference method for the Helmholtz equation using symbolic computation[J]. Int. J. Comput. Engrg. Sci., 2003, 4:121-144.[31] 刘佳勇. 二维大波数Helmholtz方程的九点差分格式[D]. 重庆大学, 2015.[32] Nabavia M, Siddiqui M, Dargahi J. A new 9-point sixth-order accurate compact finite-difference method for the Helmholtz equation[J]. J. Sound Vibrat., 2007, 307:972-982.[33] Operto S, Virieux J, Amestoy P, L'Excellent J, Giraud L, Ali H. 3D finite-difference frequencydomain modeling of visco-acoustic wave propagation using a massively parallel direct solver:A feasibility study[J]. Geophysics, 2007, 72:SM195-SM211.[34] Shin C, Sohn H. A frequency-space 2-D scalar wave extrapolator using extended 25-point finitedifference operator[J]. Geophysics, 1998, 63:289-296.[35] Singer I, Turkel E. Sixth-order accurate finite difference schemes for the Helmholtz equation[J]. J. Comput. Acoust., 2006, 14:339-351.[36] Singer I, Turkel E. High-order finite difference methods for the Helmholtz equation[J]. Comput. Methods Appl. Mech. Engrg., 1998, 163:343-358.[37] Sutmann G. Compact finite difference schemes of sixth order for the Helmholtz equation[J]. J. Comput. Appl. Math., 2007, 203:15-31.[38] Su X, Feng X, Li Z. Fourth-order compact schemes for Helmholtz equations with piecewise wave numbers in the polar coordinates[J]. J. Comput. Math., 2016, 34(5):499-510.[39] Stephane O, Jean V, Patrick A, Jean-Yves E, Giraud G, Hafedh B. 3D finite-difference frequencydomain modeling of visco-acoustic wave propagation using a massively parallel direct solver:A feasibility study[J]. Geophysics, 2007, 72:SM195-SM211.[40] Turkel E, Gordon D, Gordon R, Tsynkov S. Compact 2D and 3D sixth order schemes for the Helmholtz equation with variable wave number[J]. J. Comput. Phys., 2013, 232:272-287.[41] Wang K, Wong Y S. Pollution-free finite difference schemes for non-homogeneous Helmholtz equation[J]. Int. J. Numer. Anal. Model., 2014, 11:787-815.[42] Wang K, Wong Y S, Deng J. Efficient and accurate numerical solutions for Helmholtz equation in polar and spherical coordinates[J]. Commun. Comput. Phys., 2015, 17:779-807.[43] Wang K, Wong Y S. 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.[44] Wang K, Wong Y, Huang J. Solving Helmholtz equation at high wave numbers in exterior domains[J]. Appl. Math. Comput., 2017, 298:221-235.[45] Wang K, Wong Y S, Huang J. Analysis of pollution-free approaches for multi-dimensional Helmholtz equations[J]. submitted.[46] Wong Y S, Li G. Exact finite difference schemes for solving Helmholtz equation at any wavenumber[J]. Int. J. Numer. Anal. Model. Ser., B 2011, 2:91-108.[47] Wu T, Chen Z. A dispersion minimizing subgridding finite difference scheme for the Helmholtz equation with PML[J]. J. Comput. Appl. Math., 2014, 267:82-95.[48] Wu T, Chen Z, Chen J. Optimal 25-point finite-difference subgridding techniques for the 2D Helmholtz equation[J]. Math. Probl. Eng., 2016, Art. ID 1719846, 16 pp.[49] Wu T. A dispersion minimizing compact finite difference scheme for the 2D Helmholtz equation[J]. J. Comput. Appl. Math., 2017, 311:497-512.[50] Zhang W, Dai Y. Finite-difference solution of the Helmholtz equation based on two domain decomposition algorithms[J]. J. Appl. Math. Phys., 2013, 1:18-24.[51] Babuaska I, Sauter S. Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers[J]. SIAM J. Numer. Anal., 1997, 34(6):2392-2423.[52] Burman E, Wu H, Zhu L. Linear continuous interior penalty finite element method for Helmholtz equation with high wave number:one-dimensional analysis[J]. Numer Methods Partial Differential Eq., 2016, 32:1378-1410.[53] Chen H, Lu P, Xu X. A hybridizable discontinuous Galerkin method for the Helmholtz equation with high wave number[J]. SIAM J. Numer. Anal., 2013, 51(4):2166-2188.[54] Chen H, Lu P, Xu X. A robust multilevel method for hybridizable discontinuous Galerkin method for the Helmholtz equation[J]. J. Comput. Phys., 2014, 264:133-151.[55] Du Y, 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.[56] Feng X, Wu H. hp-discontinuous Galerkin methods for the Helmholtz equation with large wave number[J]. Math. Comp., 2011, 80(276):1997-2024.[57] Chen W, Liu Y, Xu X. A robust domain decomposition method for the Helmholtz equation with high wave number[J]. ESAIM Math. Model. Numer. Anal., 2016, 50(3):921-944.[58] Harari I, Hughes T J R. Finite element method for the Helmholtz equation in an exterior domain:model problems[J]. Comp. Methods Appl. Mech. Eng., 1991, 87:59-96.[59] 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., 2014, 34(3):1266-1288.[60] Zhu L, 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.[61] Zheng E, Ma F, Wang Y. A least-squares FEM for the direct and inverse rectangular cavity scattering problem[J], Math. Probl. Eng., 2015, Art. ID 524345, 10 pp.[62] Han H, Huang Z. A tailored finite point method for the Helmholtz equation with high wave numbers in heterogenous medium[J]. J. Comput. Math., 2008, 26(5):728-739.[63] Huang Z, Yang X. Tailored finite cell method for solving Helmholtz equation in layered Heterogeneous medium[J]. J. Comput. Math., 2012, 30:381-391.[64] Hu Q, Yuan L. A weighted variational formulation based on plane wave basis for discretization of Helmholtz equations[J]. Int. J. Numer. Anal. Model., 2014, 11(3):587-607.[65] Li S, Xiang S. Convergence analysis of a coupled method for Helmholtz equation[J]. Complex Var. Elliptic Equ., 2014, 59(4):484-503.[66] Zheng C. Gaussian beam approach for the boundary value problem of high frequency Helmholtz equation[J]. Commun. Math. Sci., 2010, 8:1041-1066.[67] Zheng C. Global geometrical optics approximation to the high frequency Helmholtz equation with discontinuous media[J]. Commun. Math. Sci., 2015, 13:1949-1974.[68] Chen Z, Cheng D, Feng W, Wu T, Yang H. A multigrid-based preconditioned Krylov subspace method for the Helmholtz equation with PML[J]. J. Math. Anal. Appl., 2011, 383(2):522-540.[69] Chen H, Wu H, Xu X. Multilevel preconditioner with stable coarse grid corrections for the Helmholtz equation[J]. SIAM J. Sci. Comput., 2015, 37(1):A221-A244.[70] Ito K, Qiao Z, Toivanen J. A domain decomposition solver for acoustic scattering by elastic objects in layered media[J]. J. Comput. Phys., 2008, 227(19):8685-8698.[71] Yuan L, Hu Q, An H. Parallel preconditioners for plane wave Helmholtz and Maxwell systems with large wave numbers[J]. Int. J. Numer. Anal. Model., 2016, 13(5):802-819.[72] Engquist B, Ying L. Sweeping preconditioner for the Helmholtz equation:Hierarchical matrix representation[J]. Commun. Pure Appl. Math., 2011, LXIV:697-735.[73] 冷伟. 求解Helmholtz方程的快速算法[J]. 工程数学学报, 2015, 5:726-742.[74] Hu Q, Zhang H. Substructuring preconditioners for the systems arising from plane wave discretization of Helmholtz equations[J]. SIAM J. Sci. Comput., 2016, 38(4):A2232-A2261.[75] Yuan L, Hu Q. A solver for Helmholtz system generated by the discretization of wave shape functions[J]. Adv. Appl. Math. Mech., 2013, 5(6):791-808.[76] Li S, Xiang S, Xian J. A fast hybrid Galerkin method for high-frequency acoustic scattering[J]. Appl. Anal., 2017, 96(10):1698-1712.[77] Zhao M, Qiao Z, Tang T. A fast high order method for electromagnetic scattering by large open cavities[J]. J. Comput. Math., 2011, 29(3):287-304. |
