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Advances in iterative methods and preconditioners for the Helmholtz equation. (English) Zbl 1158.65078
The Helmholtz equation \(\nabla ^{2}u(\mathbf{x})+\kappa ^{2}u(\mathbf{x})=h( \mathbf{x}),\) where \(\nabla ^{2}\) is the Laplacian, \(\kappa \) is the wave number, \(h\) is a forcing function and \(u\) is the amplitude, finds applications in many important fields including aeroacoustics, under-water acoustics, seismic inversion and electromagnetics. Therefore computation of its solutions in a two or three dimensional domain is important. The linear system arising from a discretization of the Helmholtz equation is typically characterized by indefiniteness of the (real part of the) eigenvalues of the corresponding coefficient matrix, and hence the corresponding iterative scheme may encounter convergence problems.
This paper reviews and highlights some recent advances in iterative methods for the Helmholtz equation. In particular, the author focuses on the Krylov subspace methods and the shifted Laplacian preconditioner. Some theories behind the shifted Laplacian preconditioner are given, and numerical results are presented for realistic problems. There are 142 references listed in this paper which include several recent surveys on similar subjects. The emphasis is on engineering computation, however, and the reader who is interested in the theoretical aspects is therefore encouraged to look for additional sources as well.

65N06 Finite difference methods for boundary value problems involving PDEs
65F10 Iterative numerical methods for linear systems
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
65F35 Numerical computation of matrix norms, conditioning, scaling
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
Full Text: DOI
[1] Abarbanel S, Gottlieb D (1997) A mathematical analysis of the PML method. J Comput Phys 134:357–363 · Zbl 0887.65122
[2] Abarbanel S, Gottlieb D (1998) On the construction and analysis of absorbing layers in CEM. Appl Numer Math 27:331–340 · Zbl 0924.35160
[3] 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 · Zbl 0474.76082
[4] Arnoldi WE (1951) The principle of minimized iterations in the solution of the matrix eigenvalue problem. Q Appl Math 9:17–29 · Zbl 0042.12801
[5] 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 · Zbl 0956.65095
[6] 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 · Zbl 0974.76042
[7] 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 · Zbl 0974.76043
[8] 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 · Zbl 0716.35036
[9] 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 · Zbl 0479.65056
[10] Bayliss A, Goldstein CI, Turkel E (1983) An iterative method for Helmholtz equation. J Comput Phys 49:443–457 · Zbl 0524.65068
[11] 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 · Zbl 0596.76092
[12] Bayliss A, Goldstein CI, Turkel E (1985) On accuracy conditions for the numerical computation of waves. J Comput Phys 59:396–404 · Zbl 0647.65072
[13] Benamou JD, Despres B (1997) Domain decomposition method for the Helmholtz equation and related optimal control problems. J Comput Phys 136:62–88 · Zbl 0884.65118
[14] Benzi M, Haws JC, Tuma M (2000) Preconditioning highly indefinite and nonsymmetric matrices. SIAM J Sci Comput 22:1333–1353 · Zbl 0985.65036
[15] Berenger JP (1994) A perfectly matched layer for the absorption of electromagnetic waves. J Comput Phys 114:185–200 · Zbl 0814.65129
[16] Berenger JP (1996) Three-dimensional perfectly matched layer for the absorption of electromagnetic waves. J Comput Phys 127:363–379 · Zbl 0862.65080
[17] Berkhout AJ (1982) Seismic migration: imaging of acoustic energy by wave field extrapolation. Elsevier, Amsterdam
[18] Bollöffer M (2004) A robust and efficient ILU that incorporates the growth of the inverse triangular factors. SIAM J Sci Comput 25:86–103 · Zbl 1038.65021
[19] Bourgeois A, Bourget M, Lailly P, Poulet M, Ricarte P, Versteeg R (1991) Marmousi, model and data. In: Marmousi experience, pp 5–16
[20] 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
[21] Brandt A (1977) Multi–level adaptive solutions to boundary–value problems. Math Comput 31:333–390 · Zbl 0373.65054
[22] Brandt A (2002) Multigrid techniques: 1984 guide with applications to fluid dynamics. Technical Report GMD-Studie 85, GMD Sankt Augustine, Germany · Zbl 0581.76033
[23] Brandt A, Livshits I (1997) Wave-ray multigrid methods for standing wave equations. Electr Trans Numer Anal 6:162–181 · Zbl 0891.65127
[24] Brandt A, Ta’asan S (1986) Multigrid method for nearly singular and slightly indefinite problems. In: Proc EMG’85 Cologne, 1986, pp 99–121
[25] Brezinzky C, Zaglia MR (1995) Look-ahead in bi-cgstab and other product methods for linear systems. BIT 35:169–201 · Zbl 0831.65032
[26] Briggs WL (1988) A multigrid tutorial. SIAM, Philadelphia
[27] Chow E, Saad Y (1997) ILUS: an incomplete LU factorization for matrices in sparse skyline format. Int J Numer Methods Fluids 25:739–749 · Zbl 0896.76037
[28] Clayton R, Engquist B (1977) Absorbing boundary conditions for acoustic and elastic wave equations. Bull Seis Soc Am 67(6):1529–1540
[29] Colloni F, Ghanemi S, Joly P (1998) Domain decomposition methods for harmonic wave propagation: a general presentation. Technical Report, INRIA RR-3473
[30] Colton D, Kress R (1983) Integral equation methods in scattering theory. Willey, New York · Zbl 0522.35001
[31] Colton D, Kress R (1998) Inverse matrix and electromagnetic scattering theory. Springer, Berlin · Zbl 0893.35138
[32] D’Azevedo EF, Forsyth FA, Tang WP (1992) Towards a cost effective ILU preconditioner with high level fill. BIT 31:442–463 · Zbl 0761.65017
[33] Dendy J Jr (1983) Blackbox multigrid for nonsymmetric problems. Appl Math Comput 13:261–283 · Zbl 0533.65063
[34] 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 · Zbl 0957.65098
[35] de Zeeuw PM (1990) Matrix-dependent prolongations and restrictions in a blackbox multigrid solver. J Comput Appl Math 33:1–27 · Zbl 0717.65099
[36] de Zeeuw PM (1996) Development of semi-coarsening techniques. Appl Numer Math 19:433–465 · Zbl 0852.65114
[37] Drespes B (1990) Domain decomposition method and Helmholtz problems. In: Cohen G, Halpern L, Joly P (eds) Mathematical and numerical aspects of wave propagation phenomena. SIAM, Philadelphia, pp 42–51
[38] Elman HC (1986) A stability analysis of incomplete LU factorizations. Math Comput 47:191–217 · Zbl 0621.65024
[39] 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 · Zbl 1004.65134
[40] Engquist B, Majda A (1977) Absorbing boundary conditions for the numerical simulation of waves. Math Comput 31:629–651 · Zbl 0367.65051
[41] Erlangga YA, Vuik C, Oosterlee CW (2004) On a class of preconditioners for solving the Helmholtz equation. Appl Numer Math 50:409–425 · Zbl 1051.65101
[42] 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
[43] Erlangga YA, Oosterlee CW, Vuik C (2006) A novel multigrid-based preconditioner for the heterogeneous Helmholtz equation. SIAM J Sci Comput 27:1471–1492 · Zbl 1095.65109
[44] 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 · Zbl 1094.65041
[45] 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
[46] Fan K (1960) Note in M-matrices. Q J Math Oxford Ser 2 11:43–49 · Zbl 0104.01203
[47] 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 · Zbl 0965.65133
[48] Fish J, Qu Y (2000) Global-basis two-level method for indefinite systems. Int J Numer Methods Eng 49:439–460 · Zbl 0980.74060
[49] 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 · Zbl 0980.74060
[50] Fletcher R (1975) Conjugate gradient methods for indefinite systems. In: Watson GA (ed) Proc the 1974 Dundee biennial conf on numerical analysis, pp 73–89
[51] Frank J, Vuik C (2001) On the construction of deflation-based preconditioners. SIAM J Sci Comput 23:442–462 · Zbl 0997.65072
[52] Freund RW (1992) Conjugate gradient-type methods for linear systems with complex symmetric coefficient matrices. SIAM J Sci Stat Comput 13(1):425–448 · Zbl 0761.65018
[53] Freund RW (1997) Preconditioning of symmetric but highly indefinite linear systems. In: Sydow A (ed) 15th IMACS world congress on scientific computation modelling and applied mathematics, vol 2. Numerical mathematics, pp 551–556
[54] Freund RW, Nachtigal NM (1991) QMR: A quasi minimum residual method for non-Hermitian linear systems. Numer Math 60:315–339 · Zbl 0754.65034
[55] Gander MJ, Nataf F (2000) AILU: a preconditioner based on the analytical factorization of the elliptical operator. Numer Linear Algebra Appl 7:543–567 · Zbl 1051.65054
[56] Gander MJ, Nataf F (2001) AILU for Helmholtz problems: a new preconditioner based on the analytic parabolic factorization. J Comput Acoust 9:1499–1509 · Zbl 1360.76181
[57] Gander MJ, Nataf F (2005) An incomplete LU preconditioner for problems in acoustics. J Comput Acoust 13:455–476 · Zbl 1189.76362
[58] George A, Liu JW (1981) Computer solution of large sparse positive definite systems. Prentice-Hall, Englewood Cliffs · Zbl 0516.65010
[59] 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
[60] Ghosh-Roy DN, Couchman LS (2002) Inverse problems and inverse scattering of plane waves. Academic, London
[61] 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
[62] Gozani J, Nachshon A, Turkel E (1984) Conjugate gradient coupled with multigrid for an indefinite problem. In: Advances in comput methods for PDEs V, pp 425–427
[63] Greenbaum A (1997) Iterative methods for solving linear systems. SIAM, Philadelphia · Zbl 0883.65022
[64] Grote MJ, Huckel T (1997) Parallel preconditioning with sparse approximate inverses. SIAM J Sci Comput 18:838–853 · Zbl 0872.65031
[65] 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 · Zbl 0961.65025
[66] 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
[67] Hackbusch W (2003) Multi-grid methods and applications. Springer, Berlin · Zbl 0595.65106
[68] Hadley GR (2006) A complex Jacobi iterative method for the indefinite Helmholtz equation. J Comput Phys 203:358–370 · Zbl 1069.65110
[69] Harari I (2006) A survey of finite element methods for time-harmonic acoustics. Comput Methods Appl Mech Eng 195:1594–1607 · Zbl 1122.76056
[70] Harari I, Turkel E (1995) Accurate finite difference methods for time-harmonic wave propagation. J Comput Phys 119:252–270 · Zbl 0848.65072
[71] Heikkola E, Rossi T, Toivanen J (2000) A parallel fictitious domain decomposition method for the three-dimensional Helmholtz equation. Technical Report No B 9/2000, Dept Math Info Tech, Univ Jÿvaskÿla · Zbl 1035.65126
[72] Hestenes MR, Stiefel E (1952) Methods of conjugate gradients for solving linear systems. J Res Nat Bur Stand 49:409–435 · Zbl 0048.09901
[73] 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 · Zbl 0851.73062
[74] 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 · Zbl 0838.65108
[75] 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 · Zbl 0884.65104
[76] 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
[77] 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 · Zbl 0505.65048
[78] Kim S (1994) A parallezable iterative procedure for the Helmholtz equation. Appl Numer Math 14:435–449 · Zbl 0805.65100
[79] Kim S (1995) Parallel multidomain iterative algorithms for the Helmholtz wave equation. Appl Numer Math 17:411–429 · Zbl 0838.65119
[80] Kim S (1998) Domain decomposition iterative procedures for solving scalar waves in the frequency domain. Numer Math 79:231–259 · Zbl 0926.65132
[81] Kononov AV, Riyanti CD, de Leeuw SW, Vuik C, Oosterlee CW (2006) Numerical performance of parallel solution of heterogeneous 2d Helmholtz equation. In: Wesseling P, Oosterlee CW, Hemker P (eds) Proceedings of the 8th European multigrid conference, TU Delft
[82] Laird AL, Giles MB (2002) Preconditioned iterative solution of the 2D Helmholtz equation. Technical Report NA 02-12, Comp Lab, Oxford Univ
[83] 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
[84] Lanczos C (1952) Solution of systems of linear equations by minimized iterations. J Res Nat Bur Stand 49:33–53
[85] Larsson E (1999) Domain decomposition method for the Helmholtz equation in a multilayer domain. SIAM J Sci Comput 20:1713–1731 · Zbl 0936.65140
[86] Lee B, Manteuffel TA, McCormick SF, Ruge J (2000) First-order system least-squares for the Helmholtz equation. SIAM J Sci Comput 21:1927–1949 · Zbl 0957.65097
[87] Lele SK (1992) Compact finite difference schemes with spectral-like resolution. J Comput Phys 103(1):16–42 · Zbl 0759.65006
[88] Lynch RE, Rice JR (1980) A high-order difference method for differential equations. Math Comput 34(150):333–372 · Zbl 0424.65037
[89] Made MMM (2001) Incomplete factorization-based preconditionings for solving the Helmholtz equation. Int J Numer Methods Eng 50:1077–1101 · Zbl 0977.65102
[90] Manteuffel TA, Parter SV (1990) Preconditioning and boundary conditions. SIAM J Numer Anal 27(3):656–694 · Zbl 0713.65064
[91] 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 · Zbl 0349.65020
[92] 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 · Zbl 0472.65028
[93] Morgan RB (1995) A restarted GMRES method augmented with eigenvectors. SIAM J Matrix Anal Appl 16:1154–1171 · Zbl 0836.65050
[94] Nicolaides RA (1987) Deflation of conjugate gradients with applications to boundary value problems. SIAM J Numer Anal 24:355–365 · Zbl 0624.65028
[95] Oosterlee CW (1995) The convergence of parallel multiblock multigrid methods. Appl Numer Math 19:115–128 · Zbl 0853.65131
[96] 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 · Zbl 0913.65109
[97] Otto K, Larsson E (1999) Iterative solution of the Helmholtz equation by a second order method. SIAM J Matrix Anal Appl 21:209–229 · Zbl 0942.65119
[98] Plessix RE, Mulder WA (2004) Separation-of-variables as a preconditioner for an iterative Helmholtz solver. Appl Numer Math 44:385–400 · Zbl 1013.65117
[99] 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
[100] Quarteroni A, Valli A (1999) Domain decomposition methods for partial differential equations. Oxford Science Publications, Oxford · Zbl 0931.65118
[101] Riyanti CD, Kononov AV, Vuik C, Oosterlee CW (2006) Parallel performance of an iterative solver for heterogeneous Helmholtz problems. In: SIAM conference on parallel processing for scientific computing, San Fransisco, CA
[102] Riyanti CD, Kononov A, Erlangga YA, Vuik C, Oosterlee CW, Plessix R-E, Mulder WA (2007) A parallel multigrid-based preconditioner for the 3D heterogeneous high-frequency Helmholtz equation. J Comput Phys 224(1):431–448 · Zbl 1120.65127
[103] Saad Y (1993) A flexible inner-outer preconditioned GMRES algorithm. SIAM J Sci Comput 14:461–469 · Zbl 0780.65022
[104] Saad Y (1994) ILUT: a dual threshold incomplete LU factorization. Numer Linear Algebra Appl 1:387–402 · Zbl 0838.65026
[105] Saad Y (2003) Iterative methods for sparse linear systems. SIAM, Philadelphia · Zbl 1031.65046
[106] 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 · Zbl 0599.65018
[107] Schenk O, Gärtner K (2004) Solving unsymmetric sparse systems of linear equations with PARDISO. J Future Gen Comput Syst 20:475–487
[108] Schenk O, Gärtner K (2006) On fast factorization pivoting methods for symmetric indefinite systems. Electron Trans Numer Anal 23:158–179 · Zbl 1112.65022
[109] Singer I, Turkel E (1998) High-order finite difference methods for the Helmholtz equation. Comput Methods Appl Mech Eng 163:343–358 · Zbl 0940.65112
[110] Singer I, Turkel E (2006) Sixth order accurate finite difference scheme for the Helmholtz equations. J Comput Acoust 14(3):339–351 · Zbl 1198.65210
[111] Smith B, Bjorstad P, Gropp W (1996) Domain decomposition: parallel multilevel methods for elliptic partial differential equations. Cambridge University Press, Cambridge · Zbl 0857.65126
[112] Sonneveld P (1989) CGS: a fast Lanczos-type solver for nonsymmetric linear systems. SIAM J Sci Stat Comput 10:36–52 · Zbl 0666.65029
[113] Strikwerda JC (1989) Finite difference schemes and partial differential equations. Wadsworth & Brooks/Cole, Pacific Groove · Zbl 0681.65064
[114] 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 · Zbl 0562.65071
[115] Susan-Resiga RF, Atassi HM (1998) A domain decomposition method for the exterior Helmholtz problem. J Comput Phys 147:388–401 · Zbl 0923.65084
[116] Szyld DB, Vogel JA (2001) A flexible quasi-minimal residual method with inexact preconditioning. SIAM J Sci Comput 23:363–380 · Zbl 0997.65062
[117] Tam CKW, Webb JC (1993) Dispersion-relation-preserving finite difference schemes for computational acoustics. J Comput Phys 107(2):262–281 · Zbl 0790.76057
[118] Tarantola A (1984) Inversion of seismic reflection data in the acoustic approximation. Geophysics 49:1259–1266
[119] 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 · Zbl 1002.76072
[120] Thole CA, Trottenberg U (1986) Basic smoothing procedures for the multigrid treatment of elliptic 3-d operators. Appl Math Comput 19:333–345 · Zbl 0612.65065
[121] Tosseli A, Widlund O (2005) Domain decomposition methods. Springer, Berlin
[122] Trottenberg U, Oosterlee C, Schüller A (2001) Multigrid. Academic, New York · Zbl 0976.65106
[123] Tsynkov S, Turkel E (2001) A Cartesian perfectly matched layer for the Helmholtz equation. In: Tourette L, Harpern L (eds) Absrobing boundaries and layers, domain decomposition methods applications to large scale computation. Springer, Berlin, pp 279–309
[124] Turkel E (2001) Numerical difficulties solving time harmonic equations. In: Multiscale computational methods in chemistry and physics. IOS, Ohmsha, pp 319–337
[125] 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
[126] 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 · Zbl 0761.65023
[127] van der Vorst HA (2003) Iterative Krylov methods for large linear systems. Cambridge University Press, New York · Zbl 1023.65027
[128] 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
[129] van der Vorst HA, Vuik C (1993) The superlinear convergence behaviour of GMRES. J Comput Appl Math 48:327–341 · Zbl 0797.65026
[130] van der Vorst HA, Vuik C (1994) GMRESR: a family for nested GMRES methods. Numer Linear Algebra Appl 1(4):369–386 · Zbl 0839.65040
[131] van Gijzen M, Erlangga YA, Vuik C (2007) Spectral analysis of the shifted Laplace precondtioner. SIAM J Sci Comput 29(5):1942–1958 · Zbl 1155.65088
[132] 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
[133] 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
[134] Vanek P, Mandel J, Brezina M (1996) Algebraic multigrid based on smoothed aggregation for second and fourth order problems. Computing 56:179–196 · Zbl 0851.65087
[135] Vanek PV, Mandel J, Brezina M (1998) Two-level algebraic multigrid for the Helmholtz problem. Contemp Math 218:349–356 · Zbl 0910.65087
[136] Vuik C, Erlangga YA, Oosterlee CW (2003) Shifted Laplace preconditioner for the Helmholtz equations. Technical Report 03-18, Dept Appl Math Anal, Delft Univ Tech, The Netherlands
[137] Waisman H, Fish J, Tuminaro RS, Shadid J (2004) The generalized global basis (GGB) methods. Int J Numer Methods Eng 61:1243–1269 · Zbl 1075.74685
[138] Washio T, Oosterlee CW (1998) Flexible multiple semicoarsening for three dimensional singularly perturbed problems. SIAM J Sci Comput 19:1646–1666 · Zbl 0913.65110
[139] Wesseling P (1992) An introduction to multigrid methods. Willey, London · Zbl 0760.65092
[140] Wienands R, Joppich W (2004) Practical Fourier analysis for multigrid methods. Chapman & Hall/CRC, London · Zbl 1062.65133
[141] Wienands R, Oosterlee CW (2001) On three-grid Fourier analysis of multigrid. SIAM J Sci Comput 23:651–671 · Zbl 0992.65137
[142] Zhou L, Walker HF (1994) Residual smoothing techniques for iterative methods. SIAM J Sci Comput 15(2):297–312 · Zbl 0802.65041
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