메뉴 건너뛰기




Volumn 15, Issue 1, 2008, Pages 37-66

Advances in iterative methods and preconditioners for the Helmholtz equation

Author keywords

[No Author keywords available]

Indexed keywords

CONVERGENCE OF NUMERICAL METHODS; HELMHOLTZ EQUATION; NUMERICAL ANALYSIS; NUMERICAL METHODS; PAPER; SURVEYS;

EID: 46349085468     PISSN: 11343060     EISSN: None     Source Type: Journal    
DOI: 10.1007/s11831-007-9013-7     Document Type: Article
Times cited : (159)

References (142)
  • 1
    • 0031188291 scopus 로고    scopus 로고
    • A mathematical analysis of the PML method
    • Abarbanel S, Gottlieb D (1997) A mathematical analysis of the PML method. J Comput Phys 134:357-363
    • (1997) J Comput Phys , vol.134 , pp. 357-363
    • Abarbanel, S.1    Gottlieb, D.2
  • 2
    • 0032139990 scopus 로고    scopus 로고
    • On the construction and analysis of absorbing layers in CEM
    • Abarbanel S, Gottlieb D (1998) On the construction and analysis of absorbing layers in CEM. Appl Numer Math 27:331-340
    • (1998) Appl Numer Math , vol.27 , pp. 331-340
    • Abarbanel, S.1    Gottlieb, D.2
  • 3
    • 0000014842 scopus 로고
    • The multi-grid method for the diffusion equation with strongly discontinuous coefficients
    • Alcouffe RE, Brandt A, Dendy JE Jr, Painter JW (1981) The multi-grid method for the diffusion equation with strongly discontinuous coefficients. SIAM J Sci Comput 2:430-454
    • (1981) SIAM J Sci Comput , vol.2 , pp. 430-454
    • Alcouffe, R.E.1    Brandt, A.2    Dendy Jr, J.E.3    Painter, J.W.4
  • 4
    • 0002807741 scopus 로고
    • The principle of minimized iterations in the solution of the matrix eigenvalue problem
    • Arnoldi WE (1951) The principle of minimized iterations in the solution of the matrix eigenvalue problem. Q Appl Math 9:17-29
    • (1951) Q Appl Math , vol.9 , pp. 17-29
    • Arnoldi, W.E.1
  • 5
    • 38849170302 scopus 로고    scopus 로고
    • Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers?
    • Babuska I, Sauter S (1997) Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers?. SIAM J Numer Anal 27:323-352
    • (1997) SIAM J Numer Anal , vol.27 , pp. 323-352
    • Babuska, I.1    Sauter, S.2
  • 6
    • 0031234255 scopus 로고    scopus 로고
    • Posteriori error estimation for finite element solutions of Helmholtz's equation. Part I: The quality of local indicators and estimators
    • Babuska I, Ihlenburg F, Strouboulis T, Gangaraj SK (1997) Posteriori error estimation for finite element solutions of Helmholtz's equation. Part I: the quality of local indicators and estimators. Int J Numer Methods Eng 40:3443-3462
    • (1997) Int J Numer Methods Eng , vol.40 , pp. 3443-3462
    • Babuska, I.1    Ihlenburg, F.2    Strouboulis, T.3    Gangaraj, S.K.4
  • 7
    • 0031270949 scopus 로고    scopus 로고
    • Posteriori error estimation for finite element solutions of Helmholtz's equation. Part II: Estimation of the pollution error
    • Babuska I, Ihlenburg F, Strouboulis T, Gangaraj SK (1997) Posteriori error estimation for finite element solutions of Helmholtz's equation. Part II: estimation of the pollution error. Int J Numer Methods Eng 40:3883-3900
    • (1997) Int J Numer Methods Eng , vol.40 , pp. 3883-3900
    • Babuska, I.1    Ihlenburg, F.2    Strouboulis, T.3    Gangaraj, S.K.4
  • 8
    • 0025413866 scopus 로고
    • Second-order absorbing boundary conditions for the wave equation: A solution for the corner problem
    • Bamberger A, Joly P, Roberts JE (1990) Second-order absorbing boundary conditions for the wave equation: a solution for the corner problem. SIAM J Numer Anal 27:323-352
    • (1990) SIAM J Numer Anal , vol.27 , pp. 323-352
    • Bamberger, A.1    Joly, P.2    Roberts, J.E.3
  • 9
    • 0000753548 scopus 로고
    • Boundary conditions for the numerical solution of elliptic equations in exterior regions
    • Bayliss A, Gunzburger M, Turkel E (1982) Boundary conditions for the numerical solution of elliptic equations in exterior regions. SIAM J Appl Math 42:430-451
    • (1982) SIAM J Appl Math , vol.42 , pp. 430-451
    • Bayliss, A.1    Gunzburger, M.2    Turkel, E.3
  • 11
    • 0021462701 scopus 로고
    • The numerical solution of the Helmholtz equation for wave propagation problems in underwater acoustics
    • Bayliss A, Goldstein CI, Turkel E (1985) The numerical solution of the Helmholtz equation for wave propagation problems in underwater acoustics. Comput Math Appl 11:655-665
    • (1985) Comput Math Appl , vol.11 , pp. 655-665
    • Bayliss, A.1    Goldstein, C.I.2    Turkel, E.3
  • 12
    • 0000350889 scopus 로고
    • On accuracy conditions for the numerical computation of waves
    • Bayliss A, Goldstein CI, Turkel E (1985) On accuracy conditions for the numerical computation of waves. J Comput Phys 59:396-404
    • (1985) J Comput Phys , vol.59 , pp. 396-404
    • Bayliss, A.1    Goldstein, C.I.2    Turkel, E.3
  • 13
    • 0031234450 scopus 로고    scopus 로고
    • Domain decomposition method for the Helmholtz equation and related optimal control problems
    • Benamou JD, Despres B (1997) Domain decomposition method for the Helmholtz equation and related optimal control problems. J Comput Phys 136:62-88
    • (1997) J Comput Phys , vol.136 , pp. 62-88
    • Benamou, J.D.1    Despres, B.2
  • 14
    • 0034884436 scopus 로고    scopus 로고
    • Preconditioning highly indefinite and nonsymmetric matrices
    • Benzi M, Haws JC, Tuma M (2000) Preconditioning highly indefinite and nonsymmetric matrices. SIAM J Sci Comput 22:1333-1353
    • (2000) SIAM J Sci Comput , vol.22 , pp. 1333-1353
    • Benzi, M.1    Haws, J.C.2    Tuma, M.3
  • 15
    • 28044459877 scopus 로고
    • A perfectly matched layer for the absorption of electromagnetic waves
    • Berenger JP (1994) A perfectly matched layer for the absorption of electromagnetic waves. J Comput Phys 114:185-200
    • (1994) J Comput Phys , vol.114 , pp. 185-200
    • Berenger, J.P.1
  • 16
    • 0030239565 scopus 로고    scopus 로고
    • Three-dimensional perfectly matched layer for the absorption of electromagnetic waves
    • Berenger JP (1996) Three-dimensional perfectly matched layer for the absorption of electromagnetic waves. J Comput Phys 127:363-379
    • (1996) J Comput Phys , vol.127 , pp. 363-379
    • Berenger, J.P.1
  • 18
    • 12144291027 scopus 로고    scopus 로고
    • A robust and efficient ILU that incorporates the growth of the inverse triangular factors
    • Bollöffer M (2004) A robust and efficient ILU that incorporates the growth of the inverse triangular factors. SIAM J Sci Comput 25:86-103
    • (2004) SIAM J Sci Comput , vol.25 , pp. 86-103
    • Bollöffer, M.1
  • 20
    • 0042402961 scopus 로고
    • Multigrid and cyclic reduction applied to the Helmholtz equation
    • Melson ND, Manteuffel TA, McCormick SF (eds)
    • Brackenridge K (1993) Multigrid and cyclic reduction applied to the Helmholtz equation. In: Melson ND, Manteuffel TA, McCormick SF (eds) Proc 6th Copper Mountain conf on multigrid methods, pp 31-41
    • (1993) Proc 6th Copper Mountain Conf on Multigrid Methods , pp. 31-41
    • Brackenridge, K.1
  • 21
    • 84966233568 scopus 로고
    • Multi-level adaptive solutions to boundary-value problems
    • Brandt A (1977) Multi-level adaptive solutions to boundary-value problems. Math Comput 31:333-390
    • (1977) Math Comput , vol.31 , pp. 333-390
    • Brandt, A.1
  • 22
    • 0004056964 scopus 로고    scopus 로고
    • Multigrid techniques: 1984 guide with applications to fluid dynamics
    • GMD Sankt Augustine, Germany
    • Brandt A (2002) Multigrid techniques: 1984 guide with applications to fluid dynamics. Technical Report GMD-Studie 85, GMD Sankt Augustine, Germany
    • (2002) Technical Report GMD-Studie , vol.85
    • Brandt, A.1
  • 23
    • 0000657234 scopus 로고    scopus 로고
    • Wave-ray multigrid methods for standing wave equations
    • Brandt A, Livshits I (1997) Wave-ray multigrid methods for standing wave equations. Electr Trans Numer Anal 6:162-181
    • (1997) Electr Trans Numer Anal , vol.6 , pp. 162-181
    • Brandt, A.1    Livshits, I.2
  • 24
    • 0009964284 scopus 로고
    • Multigrid method for nearly singular and slightly indefinite problems
    • 1986
    • Brandt A, Ta'asan S (1986) Multigrid method for nearly singular and slightly indefinite problems. In: Proc EMG'85 Cologne, 1986, pp 99-121
    • (1986) Proc EMG'85 Cologne , pp. 99-121
    • Brandt, A.1    Ta'Asan, S.2
  • 25
    • 34249762279 scopus 로고
    • Look-ahead in bi-cgstab and other product methods for linear systems
    • Brezinzky C, Zaglia MR (1995) Look-ahead in bi-cgstab and other product methods for linear systems. BIT 35:169-201
    • (1995) BIT , vol.35 , pp. 169-201
    • Brezinzky, C.1    Zaglia, M.R.2
  • 27
    • 0031251615 scopus 로고    scopus 로고
    • ILUS: An incomplete LU factorization for matrices in sparse skyline format
    • Chow E, Saad Y (1997) ILUS: an incomplete LU factorization for matrices in sparse skyline format. Int J Numer Methods Fluids 25:739-749
    • (1997) Int J Numer Methods Fluids , vol.25 , pp. 739-749
    • Chow, E.1    Saad, Y.2
  • 28
    • 0000681266 scopus 로고
    • Absorbing boundary conditions for acoustic and elastic wave equations
    • 6
    • Clayton R, Engquist B (1977) Absorbing boundary conditions for acoustic and elastic wave equations. Bull Seis Soc Am 67(6):1529-1540
    • (1977) Bull Seis Soc Am , vol.67 , pp. 1529-1540
    • Clayton, R.1    Engquist, B.2
  • 32
    • 0000256195 scopus 로고
    • Towards a cost effective ILU preconditioner with high level fill
    • D'Azevedo EF, Forsyth FA, Tang WP (1992) Towards a cost effective ILU preconditioner with high level fill. BIT 31:442-463
    • (1992) BIT , vol.31 , pp. 442-463
    • D'Azevedo, E.F.1    Forsyth, F.A.2    Tang, W.P.3
  • 33
    • 0000555209 scopus 로고
    • Blackbox multigrid for nonsymmetric problems
    • Dendy J Jr (1983) Blackbox multigrid for nonsymmetric problems. Appl Math Comput 13:261-283
    • (1983) Appl Math Comput , vol.13 , pp. 261-283
    • Dendy Jr, J.1
  • 34
    • 0344549300 scopus 로고    scopus 로고
    • Dispersion and pollution of the FEM solution for the Helmholtz equation in one, two, and three dimensions
    • Deraemaeker A, Babuska I, Bouillard P (1999) Dispersion and pollution of the FEM solution for the Helmholtz equation in one, two, and three dimensions. Int J Numer Methods Eng 46:471-499
    • (1999) Int J Numer Methods Eng , vol.46 , pp. 471-499
    • Deraemaeker, A.1    Babuska, I.2    Bouillard, P.3
  • 35
    • 0003090969 scopus 로고
    • Matrix-dependent prolongations and restrictions in a blackbox multigrid solver
    • de Zeeuw PM (1990) Matrix-dependent prolongations and restrictions in a blackbox multigrid solver. J Comput Appl Math 33:1-27
    • (1990) J Comput Appl Math , vol.33 , pp. 1-27
    • De Zeeuw, P.M.1
  • 36
    • 0039176368 scopus 로고    scopus 로고
    • Development of semi-coarsening techniques
    • de Zeeuw PM (1996) Development of semi-coarsening techniques. Appl Numer Math 19:433-465
    • (1996) Appl Numer Math , vol.19 , pp. 433-465
    • De Zeeuw, P.M.1
  • 38
    • 84966211825 scopus 로고
    • A stability analysis of incomplete LU factorizations
    • Elman HC (1986) A stability analysis of incomplete LU factorizations. Math Comput 47:191-217
    • (1986) Math Comput , vol.47 , pp. 191-217
    • Elman, H.C.1
  • 39
    • 0036344991 scopus 로고    scopus 로고
    • A multigrid method enhanced by Krylov subspace iteration for discrete Helmholtz equations
    • Elman HR, Ernst OG, O'Leary DP (2001) A multigrid method enhanced by Krylov subspace iteration for discrete Helmholtz equations. SIAM J Sci Comput 22:1291-1315
    • (2001) SIAM J Sci Comput , vol.22 , pp. 1291-1315
    • Elman, H.R.1    Ernst, O.G.2    O'Leary, D.P.3
  • 40
    • 84966208271 scopus 로고
    • Absorbing boundary conditions for the numerical simulation of waves
    • Engquist B, Majda A (1977) Absorbing boundary conditions for the numerical simulation of waves. Math Comput 31:629-651
    • (1977) Math Comput , vol.31 , pp. 629-651
    • Engquist, B.1    Majda, A.2
  • 41
    • 3142611583 scopus 로고    scopus 로고
    • On a class of preconditioners for solving the Helmholtz equation
    • Erlangga YA, Vuik C, Oosterlee CW (2004) On a class of preconditioners for solving the Helmholtz equation. Appl Numer Math 50:409-425
    • (2004) Appl Numer Math , vol.50 , pp. 409-425
    • Erlangga, Y.A.1    Vuik, C.2    Oosterlee, C.W.3
  • 42
    • 46349107308 scopus 로고    scopus 로고
    • On a robust iterative method for heterogeneous Helmholtz problems for geophysical applications
    • Erlangga YA, Vuik C, Oosterlee CW (2005) On a robust iterative method for heterogeneous Helmholtz problems for geophysical applications. Int J Numer Anal Model 2:197-208
    • (2005) Int J Numer Anal Model , vol.2 , pp. 197-208
    • Erlangga, Y.A.1    Vuik, C.2    Oosterlee, C.W.3
  • 43
    • 33746033769 scopus 로고    scopus 로고
    • A novel multigrid-based preconditioner for the heterogeneous Helmholtz equation
    • Erlangga YA, Oosterlee CW, Vuik C (2006) A novel multigrid-based preconditioner for the heterogeneous Helmholtz equation. SIAM J Sci Comput 27:1471-1492
    • (2006) SIAM J Sci Comput , vol.27 , pp. 1471-1492
    • Erlangga, Y.A.1    Oosterlee, C.W.2    Vuik, C.3
  • 44
    • 33644865436 scopus 로고    scopus 로고
    • Comparison of multigrid and incomplete LU shifted-Laplace preconditioners for the inhomogeneous Helmholtz equation
    • Erlangga YA, Vuik C, Oosterlee CW (2006) Comparison of multigrid and incomplete LU shifted-Laplace preconditioners for the inhomogeneous Helmholtz equation. Appl Numer Math 56:648-666
    • (2006) Appl Numer Math , vol.56 , pp. 648-666
    • Erlangga, Y.A.1    Vuik, C.2    Oosterlee, C.W.3
  • 45
    • 46349106854 scopus 로고    scopus 로고
    • A semicoarsening-based multigrid preconditioner for the 3D inhomogeneous Helmholtz equation
    • Wesseling P, Oosterlee CW, Hemker P (eds) September 27-30, 2005, Scheveningen, TU Delft, The Netherlands
    • Erlangga YA, Vuik C, Oosterlee CW (2006) A semicoarsening-based multigrid preconditioner for the 3D inhomogeneous Helmholtz equation. In: Wesseling P, Oosterlee CW, Hemker P (eds) Proceedings of the 8th European multigrid conference, September 27-30, 2005, Scheveningen, TU Delft, The Netherlands
    • (2006) Proceedings of the 8th European Multigrid Conference
    • Erlangga, Y.A.1    Vuik, C.2    Oosterlee, C.W.3
  • 46
    • 0013558616 scopus 로고
    • Note in M-matrices
    • Fan K (1960) Note in M-matrices. Q J Math Oxford Ser 2 11:43-49
    • (1960) Q J Math Oxford ser 2 , vol.11 , pp. 43-49
    • Fan, K.1
  • 47
    • 0034405892 scopus 로고    scopus 로고
    • A two-level domain decomposition method for the iterative solution of high frequency exterior Helmholtz problems
    • Farhat C, Macedo A, Lesoinne M (2000) A two-level domain decomposition method for the iterative solution of high frequency exterior Helmholtz problems. Numer Math 85:283-308
    • (2000) Numer Math , vol.85 , pp. 283-308
    • Farhat, C.1    MacEdo, A.2    Lesoinne, M.3
  • 48
    • 0034734473 scopus 로고    scopus 로고
    • Global-basis two-level method for indefinite systems
    • Fish J, Qu Y (2000) Global-basis two-level method for indefinite systems. Int J Numer Methods Eng 49:439-460
    • (2000) Int J Numer Methods Eng , vol.49 , pp. 439-460
    • Fish, J.1    Qu, Y.2
  • 49
    • 0034734473 scopus 로고    scopus 로고
    • Global-basis two-level method for indefinite systems. Part I: Convergence studies
    • Fish J, Qu Y (2000) Global-basis two-level method for indefinite systems. Part I: convergence studies. Int J Numer Methods Eng 49:461-478
    • (2000) Int J Numer Methods Eng , vol.49 , pp. 461-478
    • Fish, J.1    Qu, Y.2
  • 51
    • 0036227210 scopus 로고    scopus 로고
    • On the construction of deflation-based preconditioners
    • Frank J, Vuik C (2001) On the construction of deflation-based preconditioners. SIAM J Sci Comput 23:442-462
    • (2001) SIAM J Sci Comput , vol.23 , pp. 442-462
    • Frank, J.1    Vuik, C.2
  • 52
    • 0001027363 scopus 로고
    • Conjugate gradient-type methods for linear systems with complex symmetric coefficient matrices
    • 1
    • Freund RW (1992) Conjugate gradient-type methods for linear systems with complex symmetric coefficient matrices. SIAM J Sci Stat Comput 13(1):425-448
    • (1992) SIAM J Sci Stat Comput , vol.13 , pp. 425-448
    • Freund, R.W.1
  • 54
    • 25444452938 scopus 로고
    • QMR: A quasi minimum residual method for non-Hermitian linear systems
    • Freund RW, Nachtigal NM (1991) QMR: A quasi minimum residual method for non-Hermitian linear systems. Numer Math 60:315-339
    • (1991) Numer Math , vol.60 , pp. 315-339
    • Freund, R.W.1    Nachtigal, N.M.2
  • 55
    • 0034368170 scopus 로고    scopus 로고
    • AILU: A preconditioner based on the analytical factorization of the elliptical operator
    • Gander MJ, Nataf F (2000) AILU: a preconditioner based on the analytical factorization of the elliptical operator. Numer Linear Algebra Appl 7:543-567
    • (2000) Numer Linear Algebra Appl , vol.7 , pp. 543-567
    • Gander, M.J.1    Nataf, F.2
  • 56
    • 0035567809 scopus 로고    scopus 로고
    • AILU for Helmholtz problems: A new preconditioner based on the analytic parabolic factorization
    • Gander MJ, Nataf F (2001) AILU for Helmholtz problems: a new preconditioner based on the analytic parabolic factorization. J Comput Acoust 9:1499-1509
    • (2001) J Comput Acoust , vol.9 , pp. 1499-1509
    • Gander, M.J.1    Nataf, F.2
  • 57
    • 25644431975 scopus 로고    scopus 로고
    • An incomplete LU preconditioner for problems in acoustics
    • Gander MJ, Nataf F (2005) An incomplete LU preconditioner for problems in acoustics. J Comput Acoust 13:455-476
    • (2005) J Comput Acoust , vol.13 , pp. 455-476
    • Gander, M.J.1    Nataf, F.2
  • 59
    • 0012790002 scopus 로고    scopus 로고
    • A domain decomposition method for Helmholtz scattering problems
    • Bjørstad, Espedal, Keyes, (eds)
    • Ghanemi S (1998) A domain decomposition method for Helmholtz scattering problems. In: Bjørstad, Espedal, Keyes, (eds) The ninth intl conf on domain decomposition methods, pp 105-112
    • (1998) The Ninth Intl Conf on Domain Decomposition Methods , pp. 105-112
    • Ghanemi, S.1
  • 61
    • 0010033127 scopus 로고
    • Multigrid preconditioners applied to the iterative methods of singularly perturbed elliptic boundary value and scattering problems
    • Springer Berlin
    • Goldstein CI (1986) Multigrid preconditioners applied to the iterative methods of singularly perturbed elliptic boundary value and scattering problems. In: Innovative numerical methods in engineering. Springer, Berlin, pp 97-102
    • (1986) Innovative Numerical Methods in Engineering , pp. 97-102
    • Goldstein, C.I.1
  • 64
    • 0031129197 scopus 로고    scopus 로고
    • Parallel preconditioning with sparse approximate inverses
    • Grote MJ, Huckel T (1997) Parallel preconditioning with sparse approximate inverses. SIAM J Sci Comput 18:838-853
    • (1997) SIAM J Sci Comput , vol.18 , pp. 838-853
    • Grote, M.J.1    Huckel, T.2
  • 65
    • 0034144748 scopus 로고    scopus 로고
    • Look-ahead procedures for Lanczos-type product methods based on three-term recurrences
    • Gutknecht MH, Ressel KJ (2000) Look-ahead procedures for Lanczos-type product methods based on three-term recurrences. SIAM J Matrix Anal Appl 21:1051-1078
    • (2000) SIAM J Matrix Anal Appl , vol.21 , pp. 1051-1078
    • Gutknecht, M.H.1    Ressel, K.J.2
  • 66
    • 46349088116 scopus 로고
    • A fast iterative method for solving Helmholtz's equation in a general region
    • Advance Publications London
    • Hackbusch W (1978) A fast iterative method for solving Helmholtz's equation in a general region. In: Schumman U (ed) Fast elliptic solvers. Advance Publications, London, pp 112-124
    • (1978) Fast Elliptic Solvers , pp. 112-124
    • Hackbusch, W.1    Schumman, U.2
  • 68
    • 10844286559 scopus 로고    scopus 로고
    • A complex Jacobi iterative method for the indefinite Helmholtz equation
    • Hadley GR (2006) A complex Jacobi iterative method for the indefinite Helmholtz equation. J Comput Phys 203:358-370
    • (2006) J Comput Phys , vol.203 , pp. 358-370
    • Hadley, G.R.1
  • 69
    • 30944443982 scopus 로고    scopus 로고
    • A survey of finite element methods for time-harmonic acoustics
    • Harari I (2006) A survey of finite element methods for time-harmonic acoustics. Comput Methods Appl Mech Eng 195:1594-1607
    • (2006) Comput Methods Appl Mech Eng , vol.195 , pp. 1594-1607
    • Harari, I.1
  • 70
    • 28144432041 scopus 로고
    • Accurate finite difference methods for time-harmonic wave propagation
    • Harari I, Turkel E (1995) Accurate finite difference methods for time-harmonic wave propagation. J Comput Phys 119:252-270
    • (1995) J Comput Phys , vol.119 , pp. 252-270
    • Harari, I.1    Turkel, E.2
  • 72
    • 0000135303 scopus 로고
    • Methods of conjugate gradients for solving linear systems
    • Hestenes MR, Stiefel E (1952) Methods of conjugate gradients for solving linear systems. J Res Nat Bur Stand 49:409-435
    • (1952) J Res Nat Bur Stand , vol.49 , pp. 409-435
    • Hestenes, M.R.1    Stiefel, E.2
  • 73
    • 0029395109 scopus 로고
    • Dispersion analysis and error estimation of Galerkin finite element methods for the Helmholtz equation
    • Ihlenburg F, Babuska I (1995) Dispersion analysis and error estimation of Galerkin finite element methods for the Helmholtz equation. Int J Numer Methods Eng 38:3745-3774
    • (1995) Int J Numer Methods Eng , vol.38 , pp. 3745-3774
    • Ihlenburg, F.1    Babuska, I.2
  • 74
    • 0001201384 scopus 로고
    • Finite element solution of the Helmholtz equation with high wave number. Part I: The h-version of the FEM
    • 9
    • Ihlenburg F, Babuska I (1995) Finite element solution of the Helmholtz equation with high wave number. Part I: the h-version of the FEM. Comput Math Appl 30(9):9-37
    • (1995) Comput Math Appl , vol.30 , pp. 9-37
    • Ihlenburg, F.1    Babuska, I.2
  • 75
    • 0001299857 scopus 로고    scopus 로고
    • Finite element solution of the Helmholtz equation with high wave number. Part II: The hp-version of the FEM
    • Ihlenburg F, Babuska I (1997) Finite element solution of the Helmholtz equation with high wave number. Part II: the hp-version of the FEM. SIAM J Numer Anal 34:315-358
    • (1997) SIAM J Numer Anal , vol.34 , pp. 315-358
    • Ihlenburg, F.1    Babuska, I.2
  • 76
    • 0029750627 scopus 로고    scopus 로고
    • An optimal 9-point, finite difference, frequency space, 2-D scalar wave extrapolator
    • 2
    • Jo C-H, Shin C, Suh JH (1996) An optimal 9-point, finite difference, frequency space, 2-D scalar wave extrapolator. Geophysics 61(2):529-537
    • (1996) Geophysics , vol.61 , pp. 529-537
    • Jo, C.-H.1    Shin, C.2    Suh, J.H.3
  • 77
    • 0000876881 scopus 로고
    • Analysis and comparison of relaxation schemes in robust multigrid and preconditioned conjugate gradient methods
    • Hackbusch W, Trottenberg U (eds) Multigrid methods
    • Kettler R (1982) Analysis and comparison of relaxation schemes in robust multigrid and preconditioned conjugate gradient methods. In: Hackbusch W, Trottenberg U (eds) Multigrid methods. Lecture notes in mathematics, vol 960, pp 502-534
    • (1982) Lecture Notes in Mathematics , vol.960 , pp. 502-534
    • Kettler, R.1
  • 78
    • 38149146331 scopus 로고
    • A parallezable iterative procedure for the Helmholtz equation
    • Kim S (1994) A parallezable iterative procedure for the Helmholtz equation. Appl Numer Math 14:435-449
    • (1994) Appl Numer Math , vol.14 , pp. 435-449
    • Kim, S.1
  • 79
    • 0012768502 scopus 로고
    • Parallel multidomain iterative algorithms for the Helmholtz wave equation
    • Kim S (1995) Parallel multidomain iterative algorithms for the Helmholtz wave equation. Appl Numer Math 17:411-429
    • (1995) Appl Numer Math , vol.17 , pp. 411-429
    • Kim, S.1
  • 80
    • 0039243579 scopus 로고    scopus 로고
    • Domain decomposition iterative procedures for solving scalar waves in the frequency domain
    • Kim S (1998) Domain decomposition iterative procedures for solving scalar waves in the frequency domain. Numer Math 79:231-259
    • (1998) Numer Math , vol.79 , pp. 231-259
    • Kim, S.1
  • 83
    • 0000094594 scopus 로고
    • An iteration method for the solution of the eigenvalue problem of linear differential and integral operators
    • Lanczos C (1950) An iteration method for the solution of the eigenvalue problem of linear differential and integral operators. J Res Nat Bur Stand 45:255-282
    • (1950) J Res Nat Bur Stand , vol.45 , pp. 255-282
    • Lanczos, C.1
  • 84
    • 0002814189 scopus 로고
    • Solution of systems of linear equations by minimized iterations
    • Lanczos C (1952) Solution of systems of linear equations by minimized iterations. J Res Nat Bur Stand 49:33-53
    • (1952) J Res Nat Bur Stand , vol.49 , pp. 33-53
    • Lanczos, C.1
  • 85
    • 0033297033 scopus 로고    scopus 로고
    • Domain decomposition method for the Helmholtz equation in a multilayer domain
    • Larsson E (1999) Domain decomposition method for the Helmholtz equation in a multilayer domain. SIAM J Sci Comput 20:1713-1731
    • (1999) SIAM J Sci Comput , vol.20 , pp. 1713-1731
    • Larsson, E.1
  • 87
    • 9144220381 scopus 로고
    • Compact finite difference schemes with spectral-like resolution
    • 1
    • Lele SK (1992) Compact finite difference schemes with spectral-like resolution. J Comput Phys 103(1):16-42
    • (1992) J Comput Phys , vol.103 , pp. 16-42
    • Lele, S.K.1
  • 88
    • 84966246071 scopus 로고
    • A high-order difference method for differential equations
    • 150
    • Lynch RE, Rice JR (1980) A high-order difference method for differential equations. Math Comput 34(150):333-372
    • (1980) Math Comput , vol.34 , pp. 333-372
    • Lynch, R.E.1    Rice, J.R.2
  • 89
    • 0035916190 scopus 로고    scopus 로고
    • Incomplete factorization-based preconditionings for solving the Helmholtz equation
    • Made MMM (2001) Incomplete factorization-based preconditionings for solving the Helmholtz equation. Int J Numer Methods Eng 50:1077-1101
    • (2001) Int J Numer Methods Eng , vol.50 , pp. 1077-1101
    • Made, M.M.M.1
  • 90
    • 0025447701 scopus 로고
    • Preconditioning and boundary conditions
    • 3
    • Manteuffel TA, Parter SV (1990) Preconditioning and boundary conditions. SIAM J Numer Anal 27(3):656-694
    • (1990) SIAM J Numer Anal , vol.27 , pp. 656-694
    • Manteuffel, T.A.1    Parter, S.V.2
  • 91
    • 84966225158 scopus 로고
    • An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix
    • 137
    • Meijerink JA, van der Vorst HA (1977) An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix. Math Comput 31(137):148-162
    • (1977) Math Comput , vol.31 , pp. 148-162
    • Meijerink, J.A.1    Van Der Vorst, H.A.2
  • 92
    • 49049148707 scopus 로고
    • Guidelines for the usage of incomplete decompositions in solving sets of linear equations as they occur in practical problems
    • Meijerink JA, van der Vorst HA (1981) Guidelines for the usage of incomplete decompositions in solving sets of linear equations as they occur in practical problems. J Comput Phys 44:134-155
    • (1981) J Comput Phys , vol.44 , pp. 134-155
    • Meijerink, J.A.1    Van Der Vorst, H.A.2
  • 93
    • 0342602167 scopus 로고
    • A restarted GMRES method augmented with eigenvectors
    • Morgan RB (1995) A restarted GMRES method augmented with eigenvectors. SIAM J Matrix Anal Appl 16:1154-1171
    • (1995) SIAM J Matrix Anal Appl , vol.16 , pp. 1154-1171
    • Morgan, R.B.1
  • 94
    • 0023331865 scopus 로고
    • Deflation of conjugate gradients with applications to boundary value problems
    • Nicolaides RA (1987) Deflation of conjugate gradients with applications to boundary value problems. SIAM J Numer Anal 24:355-365
    • (1987) SIAM J Numer Anal , vol.24 , pp. 355-365
    • Nicolaides, R.A.1
  • 95
    • 0029405501 scopus 로고
    • The convergence of parallel multiblock multigrid methods
    • Oosterlee CW (1995) The convergence of parallel multiblock multigrid methods. Appl Numer Math 19:115-128
    • (1995) Appl Numer Math , vol.19 , pp. 115-128
    • Oosterlee, C.W.1
  • 96
    • 0001927801 scopus 로고    scopus 로고
    • An evaluation of parallel multigrid as a solver and as a preconditioner for singularly perturbed problems
    • Oosterlee CW, Washio T (1998) An evaluation of parallel multigrid as a solver and as a preconditioner for singularly perturbed problems. SIAM J Sci Comput 19:87-110
    • (1998) SIAM J Sci Comput , vol.19 , pp. 87-110
    • Oosterlee, C.W.1    Washio, T.2
  • 97
    • 0033233449 scopus 로고    scopus 로고
    • Iterative solution of the Helmholtz equation by a second order method
    • Otto K, Larsson E (1999) Iterative solution of the Helmholtz equation by a second order method. SIAM J Matrix Anal Appl 21:209-229
    • (1999) SIAM J Matrix Anal Appl , vol.21 , pp. 209-229
    • Otto, K.1    Larsson, E.2
  • 98
    • 0037296994 scopus 로고    scopus 로고
    • Separation-of-variables as a preconditioner for an iterative Helmholtz solver
    • Plessix RE, Mulder WA (2004) Separation-of-variables as a preconditioner for an iterative Helmholtz solver. Appl Numer Math 44:385-400
    • (2004) Appl Numer Math , vol.44 , pp. 385-400
    • Plessix, R.E.1    Mulder, W.A.2
  • 99
    • 0025199116 scopus 로고
    • Inverse theory applied to multi-source cross-hole tomography. Part 1: Acoustic wave-equation method
    • Pratt RG, Worthington MH (1990) Inverse theory applied to multi-source cross-hole tomography. Part 1: acoustic wave-equation method. Geophys Prosp 38:287-310
    • (1990) Geophys Prosp , vol.38 , pp. 287-310
    • Pratt, R.G.1    Worthington, M.H.2
  • 103
    • 0000954680 scopus 로고
    • A flexible inner-outer preconditioned GMRES algorithm
    • Saad Y (1993) A flexible inner-outer preconditioned GMRES algorithm. SIAM J Sci Comput 14:461-469
    • (1993) SIAM J Sci Comput , vol.14 , pp. 461-469
    • Saad, Y.1
  • 104
    • 84985408358 scopus 로고
    • ILUT: A dual threshold incomplete LU factorization
    • Saad Y (1994) ILUT: a dual threshold incomplete LU factorization. Numer Linear Algebra Appl 1:387-402
    • (1994) Numer Linear Algebra Appl , vol.1 , pp. 387-402
    • Saad, Y.1
  • 106
    • 0000048673 scopus 로고
    • GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems
    • 12
    • Saad Y, Schultz MH (1986) GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J Sci Stat Comput 7(12):856-869
    • (1986) SIAM J Sci Stat Comput , vol.7 , pp. 856-869
    • Saad, Y.1    Schultz, M.H.2
  • 107
    • 1642370513 scopus 로고    scopus 로고
    • Solving unsymmetric sparse systems of linear equations with PARDISO
    • Schenk O, Gärtner K (2004) Solving unsymmetric sparse systems of linear equations with PARDISO. J Future Gen Comput Syst 20:475-487
    • (2004) J Future Gen Comput Syst , vol.20 , pp. 475-487
    • Schenk, O.1    Gärtner, K.2
  • 108
    • 33846230007 scopus 로고    scopus 로고
    • On fast factorization pivoting methods for symmetric indefinite systems
    • Schenk O, Gärtner K (2006) On fast factorization pivoting methods for symmetric indefinite systems. Electron Trans Numer Anal 23:158-179
    • (2006) Electron Trans Numer Anal , vol.23 , pp. 158-179
    • Schenk, O.1    Gärtner, K.2
  • 109
    • 0032155919 scopus 로고    scopus 로고
    • High-order finite difference methods for the Helmholtz equation
    • Singer I, Turkel E (1998) High-order finite difference methods for the Helmholtz equation. Comput Methods Appl Mech Eng 163:343-358
    • (1998) Comput Methods Appl Mech Eng , vol.163 , pp. 343-358
    • Singer, I.1    Turkel, E.2
  • 110
    • 33749584715 scopus 로고    scopus 로고
    • Sixth order accurate finite difference scheme for the Helmholtz equations
    • 3
    • Singer I, Turkel E (2006) Sixth order accurate finite difference scheme for the Helmholtz equations. J Comput Acoust 14(3):339-351
    • (2006) J Comput Acoust , vol.14 , pp. 339-351
    • Singer, I.1    Turkel, E.2
  • 112
    • 0002716979 scopus 로고
    • CGS: A fast Lanczos-type solver for nonsymmetric linear systems
    • Sonneveld P (1989) CGS: a fast Lanczos-type solver for nonsymmetric linear systems. SIAM J Sci Stat Comput 10:36-52
    • (1989) SIAM J Sci Stat Comput , vol.10 , pp. 36-52
    • Sonneveld, P.1
  • 114
    • 0001822668 scopus 로고
    • Multigrid methods: Fundamental algorithms, model problem analysis and applications
    • Hackbusch W, Trottenberg U (eds)
    • Stüben K, Trottenberg U (1982) Multigrid methods: fundamental algorithms, model problem analysis and applications. In: Hackbusch W, Trottenberg U (eds) Lecture notes in math, vol 960, pp 1-176
    • (1982) Lecture Notes in Math , vol.960 , pp. 1-176
    • Stüben, K.1    Trottenberg, U.2
  • 115
    • 0040683523 scopus 로고    scopus 로고
    • A domain decomposition method for the exterior Helmholtz problem
    • Susan-Resiga RF, Atassi HM (1998) A domain decomposition method for the exterior Helmholtz problem. J Comput Phys 147:388-401
    • (1998) J Comput Phys , vol.147 , pp. 388-401
    • Susan-Resiga, R.F.1    Atassi, H.M.2
  • 116
    • 0036229185 scopus 로고    scopus 로고
    • A flexible quasi-minimal residual method with inexact preconditioning
    • Szyld DB, Vogel JA (2001) A flexible quasi-minimal residual method with inexact preconditioning. SIAM J Sci Comput 23:363-380
    • (2001) SIAM J Sci Comput , vol.23 , pp. 363-380
    • Szyld, D.B.1    Vogel, J.A.2
  • 117
    • 19644369292 scopus 로고
    • Dispersion-relation-preserving finite difference schemes for computational acoustics
    • 2
    • Tam CKW, Webb JC (1993) Dispersion-relation-preserving finite difference schemes for computational acoustics. J Comput Phys 107(2):262-281
    • (1993) J Comput Phys , vol.107 , pp. 262-281
    • Tam, C.K.W.1    Webb, J.C.2
  • 118
    • 0021474580 scopus 로고
    • Inversion of seismic reflection data in the acoustic approximation
    • Tarantola A (1984) Inversion of seismic reflection data in the acoustic approximation. Geophysics 49:1259-1266
    • (1984) Geophysics , vol.49 , pp. 1259-1266
    • Tarantola, A.1
  • 119
    • 0035839293 scopus 로고    scopus 로고
    • Iterative solution of large-scale acoustic scattering problems with multiple right hand-sides by a domain decomposition method with Lagrange multipliers
    • Tezaur R, Macedo A, Farhat C (2001) Iterative solution of large-scale acoustic scattering problems with multiple right hand-sides by a domain decomposition method with Lagrange multipliers. Int J Numer Methods Eng 51:1175-1193
    • (2001) Int J Numer Methods Eng , vol.51 , pp. 1175-1193
    • Tezaur, R.1    MacEdo, A.2    Farhat, C.3
  • 120
    • 0001689452 scopus 로고
    • Basic smoothing procedures for the multigrid treatment of elliptic 3-d operators
    • Thole CA, Trottenberg U (1986) Basic smoothing procedures for the multigrid treatment of elliptic 3-d operators. Appl Math Comput 19:333-345
    • (1986) Appl Math Comput , vol.19 , pp. 333-345
    • Thole, C.A.1    Trottenberg, U.2
  • 125
    • 46349086197 scopus 로고    scopus 로고
    • Preconditioning a finite element solver of the Helmholtz equation
    • Wesseling P, Oñate EO, Périaux J (eds) TU Delft
    • Turkel E, Erlangga YA (2006) Preconditioning a finite element solver of the Helmholtz equation. In: Wesseling P, Oñate EO, Périaux J (eds), Proceedings ECCOMAS CFD 2006, TU Delft
    • (2006) Proceedings ECCOMAS CFD 2006
    • Turkel, E.1    Erlangga, Y.A.2
  • 126
    • 0000005482 scopus 로고
    • Bi-CGSTAB: A fast and smoothly converging variant of BI-CG for the solution of nonsymmetric linear systems
    • 2
    • van der Vorst HA (1992) Bi-CGSTAB: a fast and smoothly converging variant of BI-CG for the solution of nonsymmetric linear systems. SIAM J Sci Stat Comput 13(2):631-644
    • (1992) SIAM J Sci Stat Comput , vol.13 , pp. 631-644
    • Van Der Vorst, H.A.1
  • 128
    • 0025401744 scopus 로고
    • A Petrov-Galerkin type method for solving Ax=b, where A is symmetric complex systems
    • 2
    • van der Vorst HA, Melissen JBM (1990) A Petrov-Galerkin type method for solving Ax=b, where A is symmetric complex systems. IEEE Trans Magn 26(2):706-708
    • (1990) IEEE Trans Magn , vol.26 , pp. 706-708
    • Van Der Vorst, H.A.1    Melissen, J.B.M.2
  • 129
    • 0027701865 scopus 로고
    • The superlinear convergence behaviour of GMRES
    • van der Vorst HA, Vuik C (1993) The superlinear convergence behaviour of GMRES. J Comput Appl Math 48:327-341
    • (1993) J Comput Appl Math , vol.48 , pp. 327-341
    • Van Der Vorst, H.A.1    Vuik, C.2
  • 131
    • 46349085492 scopus 로고    scopus 로고
    • Spectral analysis of the shifted Laplace precondtioner
    • 5
    • van Gijzen M, Erlangga YA, Vuik C (2007) Spectral analysis of the shifted Laplace precondtioner. SIAM J Sci Comput 29(5):1942-1958
    • (2007) SIAM J Sci Comput , vol.29 , pp. 1942-1958
    • Van Gijzen, M.1    Erlangga, Y.A.2    Vuik, C.3
  • 132
    • 46349091385 scopus 로고    scopus 로고
    • Extensions of the complex Jacobi iteration to simulate photonic wavelength scale components
    • Wesseling P, Oñate E, Périaux J (eds) TU Delft
    • Vandersteegen P, Bienstman P, Baets R (2006) Extensions of the complex Jacobi iteration to simulate photonic wavelength scale components. In: Wesseling P, Oñate E, Périaux J (eds) Proceedings ECCOMAS CFD 2006, TU Delft
    • (2006) Proceedings ECCOMAS CFD 2006
    • Vandersteegen, P.1    Bienstman, P.2    Baets, R.3
  • 133
    • 33646189815 scopus 로고    scopus 로고
    • Using the complex Jacobi method to simulate Kerr non-linear photonic components
    • Vandersteegen P, Maes B, Bienstman P, Baets R (2006) Using the complex Jacobi method to simulate Kerr non-linear photonic components. Opt Quantum Electron 38:35-44
    • (2006) Opt Quantum Electron , vol.38 , pp. 35-44
    • Vandersteegen, P.1    Maes, B.2    Bienstman, P.3    Baets, R.4
  • 134
    • 0030388574 scopus 로고    scopus 로고
    • Algebraic multigrid based on smoothed aggregation for second and fourth order problems
    • Vanek P, Mandel J, Brezina M (1996) Algebraic multigrid based on smoothed aggregation for second and fourth order problems. Computing 56:179-196
    • (1996) Computing , vol.56 , pp. 179-196
    • Vanek, P.1    Mandel, J.2    Brezina, M.3
  • 135
    • 0008858386 scopus 로고    scopus 로고
    • Two-level algebraic multigrid for the Helmholtz problem
    • Vanek PV, Mandel J, Brezina M (1998) Two-level algebraic multigrid for the Helmholtz problem. Contemp Math 218:349-356
    • (1998) Contemp Math , vol.218 , pp. 349-356
    • Vanek, P.V.1    Mandel, J.2    Brezina, M.3
  • 138
    • 0001927802 scopus 로고    scopus 로고
    • Flexible multiple semicoarsening for three dimensional singularly perturbed problems
    • Washio T, Oosterlee CW (1998) Flexible multiple semicoarsening for three dimensional singularly perturbed problems. SIAM J Sci Comput 19:1646-1666
    • (1998) SIAM J Sci Comput , vol.19 , pp. 1646-1666
    • Washio, T.1    Oosterlee, C.W.2
  • 141
    • 0036227289 scopus 로고    scopus 로고
    • On three-grid Fourier analysis of multigrid
    • Wienands R, Oosterlee CW (2001) On three-grid Fourier analysis of multigrid. SIAM J Sci Comput 23:651-671
    • (2001) SIAM J Sci Comput , vol.23 , pp. 651-671
    • Wienands, R.1    Oosterlee, C.W.2
  • 142
    • 0000848766 scopus 로고
    • Residual smoothing techniques for iterative methods
    • 2
    • Zhou L, Walker HF (1994) Residual smoothing techniques for iterative methods. SIAM J Sci Comput 15(2):297-312
    • (1994) SIAM J Sci Comput , vol.15 , pp. 297-312
    • Zhou, L.1    Walker, H.F.2


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.