| [1] Adams R. Sobolev Spaces. Academic Press. New York, 1975.[2] Babuška I and Sauter S A. Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers?[J]. SIAM Rev., 2000, 42:451-484.[3] Bayliss A, Goldstein C, Turkel E. An iterative method for Helmholtz equation[J]. J Comput Phys., 1983, 49:443-457.[4] Brandt A and Livshits I. Wave-ray multigrid method for standing wave equations[J]. Electron. Trans. Numer. Anal., 1997, 6:162-181.[5] 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.[6] Elman H C, Ernst O G and O'Leary D P. A multigrid method enhanced by Krylov subspace iteration for discrete Helmholtz equations[J]. SIAM J. Sci. Comput., 2001, 23:1291-1315.[7] Engquist B and Ying L. Sweeping preconditioner for the Helmholtz equation:hierarchical matrix representation[J]. Comm. Pure Appl. Math., 2011, 64:697-735.[8] Engquist B and Ying L. Sweeping preconditioner for the Helmholtz equation:moving perfectly matched layers[J]. Multiscale Model. Simul., 2011, 9:686-710.[9] Erlangga Y A. Advances in iterative methods and preconditioners for the Helmholtz equation[J]. Arch. Comput. Methods Eng., 2008, 15:37-66.[10] Erlangga Y A, Vuik C and Oosterlee C W. On a class of preconditioners for solving the Helmholtz equation[J]. Appl. Numer. Math., 2004, 50:409-425.[11] Erlangga Y A, Oosterlee C W and Vuik C. A novel multigrid based preconditioner for heterogeneous Helmholtz problems[J]. SIAM J. Sci. Comput., 2006, 27:1471-1492.[12] Ernst O G and Gander M J. Why it is difficult to solve Helmholtz problems with classical iterative methods. Numerical Analysis of Multiscale Problems, Durham LMS Symposium, Citeseer, 2010.[13] Feng X and Wu H. Discontinuous Galerkin methods for the Helmholtz equation with large wave number[J]. SIAM J. Numer. Anal., 2009, 47:2872-2896.[14] Feng X and Wu H. hp-discontinuous Galerkin methods for the Helmholtz equation with large wave number[J]. Math. Comp., 2011, 80:1997-2024.[15] Ihlenburg F. Finite Element Analysis of Acoustic Scattering. Springer-Verlag, New York, 1998.[16] Laird A, Giles M. Preconditioned iterative solution of the 2D Helmholtz equation, Technical Report NA 02-12, Comp Lab, Oxford Univ, 2002.[17] Livshits I and Brandt A. Accuracy properties of the wave-ray multigrid algorithm for Helmholtz equations[J]. SIAM J. Sci. Comput., 2006, 28:1228-1251.[18] Lu P and Xu X. A robust multilevel method for the time-harmonic Maxwell equation with high wave number[J]. SIAM J. Sci. Comput., 2016, 38:A856-A874.[19] Lu P, Wu H and Xu X. Continuous interior penalty finite element methods for the time-harmonic Maxwell equation with high wave number, submitted.[20] Melenk J M and Sauter S. Convergence analysis for finite element discretizations of the Helmholtz equation with Dirichlet-to-Neumann boundary conditions[M]. Math. Comp., 2010, 79:1871-1914.[21] Melenk J M and Sauter S. Wavenumber explicit convergence analysis for Galerkin discretizations of the Helmholtz equation[J]. SIAM J. Numer. Anal., 2011, 49:1210-1243.[22] 武海军. 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.[23] Starke G. On the Performance of Krylov Subspace Iterations as Smoothers in Multigrid Methods. manuscript, 1995.[24] Wu H. Pre-asymptotic error analysis of CIP-FEM and FEM for Helmholtz equation with high wave number. Part I:Linear version[J]. IMA J. Numer. Anal., 2014, 34:1266-1288.[25] Zhu L and Wu H. Pre-asymptotic Error Analysis of CIP-FEM and FEM for Helmholtz Equation with High Wave Number. Part Ⅱ:hp version[J]. SIAM J. Numer. Anal., 2012, 51:1828-1852.[26] Saad Y. Iterative Methods for Sparse Linear Systems. PWS, Boston, 1995. |
