Numerical Algorithms

, Volume 50, Issue 3, pp 297–380 | Cite as

Equivalent operator preconditioning for elliptic problems

  • O. Axelsson
  • J. Karátson
Original Paper


The numerical solution of linear elliptic partial differential equations most often involves a finite element or finite difference discretization. To preserve sparsity, the arising system is normally solved using an iterative solution method, commonly a preconditioned conjugate gradient method. Preconditioning is a crucial part of such a solution process. In order to enable the solution of very large-scale systems, it is desirable that the total computational cost will be of optimal order, i.e. proportional to the degrees of freedom of the approximation used, which also induces mesh independent convergence of the iteration. This paper surveys the equivalent operator approach, which has proven to provide an efficient general framework to construct such preconditioners. Hereby one first approximates the given differential operator by some simpler differential operator, and then chooses as preconditioner the discretization of this operator for the same mesh. In this survey we give a uniform presentation of this approach, including theoretical foundation and several practically important applications for both symmetric and nonsymmetric equations and systems, and some nonlinear examples in the context of Newton linearization.


Elliptic problem Conjugate gradient method Preconditioning Equivalent operators Compact operators 


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  1. 1.
    Antal, I., Karátson, J.: A mesh independent superlinear algorithm for some nonlinear nonsymmetric elliptic systems. Comput. Math. Appl. (2008, in press)Google Scholar
  2. 2.
    Ashby, S.F., Manteuffel, T.A., Saylor, P.E.: A taxonomy for conjugate gradient methods. SIAM J. Numer. Anal. 27(6), 1542–1568 (1990)MATHMathSciNetGoogle Scholar
  3. 3.
    Axelsson, O.: A generalized conjugate gradient least square method. Numer. Math. 51, 209–227 (1987)MATHMathSciNetGoogle Scholar
  4. 4.
    Axelsson, O.: Iterative Solution Methods. Cambridge University Press, Cambridge (1994)MATHGoogle Scholar
  5. 5.
    Axelsson, O.: On iterative solvers in structural mechanics; separate displacement orderings and mixed variable methods. Math. Comput. Simul. 50(1–4), 11–30 (1999)MATHMathSciNetGoogle Scholar
  6. 6.
    Axelsson, O., Barker, V.A., Neytcheva, M., Polman, B.: Solving the Stokes problem on a massively parallel computer. Math. Model. Anal. 6(1), 7–27 (2001)MATHMathSciNetGoogle Scholar
  7. 7.
    Axelsson, O., Blaheta, R., Neytcheva, M.: Preconditioning of boundary value problems using elementwise Schur complements. Technical Report 2006-048, Department of Information Technology, Uppsala University, November (2006)Google Scholar
  8. 8.
    Axelsson, O., Blaheta, R., Neytcheva, M.: A black-box generalized conjugate gradient minimum residual method based on variable preconditioners and local element approximations. TR 2007-033, Institute for Information Technology, Uppsala University, December (2007)Google Scholar
  9. 9.
    Axelsson, O., Faragó, I., Karátson, J.: Sobolev space preconditioning for Newton’s method using domain decomposition. Numer. Linear Algebr. Appl. 9, 585–598 (2002)MATHGoogle Scholar
  10. 10.
    Axelsson, O., Gololobov, S.V.: A combined method of local Green’s functions and central difference method for singularly perturbed convection-diffusion problems. J. Comput. Appl. Math. 161(2), 245–257 (2003)MATHMathSciNetGoogle Scholar
  11. 11.
    Axelsson, O., Gustafsson, I.: Iterative methods for the solution of the Navier equations of elasticity. Comput. Methods Appl. Mech. Eng. 15(2), 241–258 (1978)MATHMathSciNetGoogle Scholar
  12. 12.
    Axelsson, O., Kaporin, I.: On the sublinear and superlinear rate of convergence of conjugate gradient methods. Mathematical journey through analysis, matrix theory and scientific computation (Kent, OH, 1999). Numer. Algorithms 25(1–4), 1–22 (2000)MATHMathSciNetGoogle Scholar
  13. 13.
    Axelsson, O., Karátson, J.: Symmetric part preconditioning for the conjugate gradient method in Hilbert space. Numer. Funct. Anal. 24(5–6), 455–474 (2003)MATHGoogle Scholar
  14. 14.
    Axelsson, O., Karátson J.: Conditioning analysis of separate displacement preconditioners for some nonlinear elasticity systems. Math. Comput. Simul. 64(6), 649–668 (2004)Google Scholar
  15. 15.
    Axelsson, O., Karátson J.: Superlinearly convergent CG methods via equivalent preconditioning for nonsymmetric elliptic operators. Numer. Math. 99(2), 197–223 (2004)MATHMathSciNetGoogle Scholar
  16. 16.
    Axelsson, O., Karátson J.: Symmetric part preconditioning of the CGM for Stokes type saddle-point systems. Numer. Funct. Anal. 28(9–10), 1027–1049 (2007)MATHGoogle Scholar
  17. 17.
    Axelsson, O., Karátson J.: Mesh independent superlinear PCG rates via compact-equivalent operators. SIAM J. Numer. Anal. 45(4), 1495–1516 (2007) (electronic)MATHMathSciNetGoogle Scholar
  18. 18.
    Axelsson, O., Kolotilina, L.: Diagonally compensated reduction and related preconditioning methods. Numer. Linear Algebr. Appl. 1(2), 155–177 (1994)MATHMathSciNetGoogle Scholar
  19. 19.
    Axelsson, O., Maubach, J.: On the updating and assembly of the Hessian matrix in finite element methods. Comput. Methods Appl. Mech. Eng. 71, 41–67 (1988)MATHMathSciNetGoogle Scholar
  20. 20.
    Axelsson, O., Neytcheva, M.: Scalable algorithms for the solution of Navier’s equations of elasticity. J. Comput. Appl. Math. 63(1–3), 149–178 (1995)MATHMathSciNetGoogle Scholar
  21. 21.
    Axelsson, O., Neytcheva, M.: An iterative solution method for Schur complement systems with inexact inner solver. In: Iliev, O., Kaschiev, M., Margenov, S., Sendov, B., Vassilevski, P.S. (eds.) Recent Advances in Numerical Methods and Applications II, pp. 795–803. World Scientific, Singapore (1999)Google Scholar
  22. 22.
    Axelsson, O., Vassilevski, P.S.: Algebraic multilevel preconditioning methods I. Numer. Math. 56, 157–177 (1989)MATHMathSciNetGoogle Scholar
  23. 23.
    Axelsson, O., Vassilevski, P.S.: Algebraic multilevel preconditioning methods II. SIAM J. Numer. Anal. 27, 1569–1590 (1990)MATHMathSciNetGoogle Scholar
  24. 24.
    Axelsson, O., Vassilevski, P.S.: Variable-step multilevel preconditioning methods. I. Self-adjoint and positive definite elliptic problems. Numer. Linear Algebr. Appl. 1(1), 75–101 (1994)MATHMathSciNetGoogle Scholar
  25. 25.
    Babuška, I.: The finite element method with Lagrangian multipliers. Numer. Math. 20, 179–192 (1972/73)MathSciNetGoogle Scholar
  26. 26.
    Bängtsson, E., Neytcheva, M.: Finite element block-factorized preconditioners. Technical Reports from the Department of Information Technology, Uppsala University, No. 2007-008, March (2007)Google Scholar
  27. 27.
    Bank, R.E.: Marching algorithms for elliptic boundary value problems. II. The variable coefficient case. SIAM J. Numer. Anal. 14(5), 950–970 (1977)MATHMathSciNetGoogle Scholar
  28. 28.
    Benzi, M., Golub, G.H., Liesen, J.: Numerical solution of saddle point problems. Acta Numer. 14, 1–137 (2005)MATHMathSciNetGoogle Scholar
  29. 29.
    Blaheta, R.: Displacement decomposition—incomplete factorization preconditioning techniques for linear elasticity problems. Numer. Linear Algebr. Appl. 1(2), 107–128 (1994)MATHMathSciNetGoogle Scholar
  30. 30.
    Blaheta, R.: Multilevel Newton methods for nonlinear problems with applications to elasticity. Copernicus 940820, Technical Report. Ostrava (1997)Google Scholar
  31. 31.
    Börgers, C., Widlund, O.B.: On finite element domain imbedding methods. SIAM J. Numer. Anal. 27(4), 963–978 (1990)MATHMathSciNetGoogle Scholar
  32. 32.
    Bramble, J.H., Pasciak, J.E.: Preconditioned iterative methods for nonselfadjoint or indefinite elliptic boundary value problems. In: Unification of Finite Element Methods. North-Holland Math. Stud. vol. 94, pp. 167–184. North-Holland, Amsterdam (1984)Google Scholar
  33. 33.
    Brezinski, C., Sadok, H.: Lanczos-type algorithms for solving systems of linear equations. Appl. Numer. Math. 11(6), 443–473 (1993)MATHMathSciNetGoogle Scholar
  34. 34.
    Brezzi, F.: On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers. Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 8(R-2), 129–151 (1974)MathSciNetGoogle Scholar
  35. 35.
    Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer, New York (1991)MATHGoogle Scholar
  36. 36.
    Brezzi, F., Raviart, P.-A.: Mixed finite element methods for 4th order elliptic equations. In: Topics in Numerical Analysis III, pp. 33–56. Proc. Roy. Irish Acad. Conf., Trinity Coll., Dublin (1976)Google Scholar
  37. 37.
    Cao, W., Haynes, R.D., Trummer, M.R.: Preconditioning for a class of spectral differentiation matrices. J. Sci. Comput. 24(3), 343–371 (2005)MATHMathSciNetGoogle Scholar
  38. 38.
    Carey, G.F., Jiang, B.-N.: Nonlinear preconditioned conjugate gradient and least-squares finite elements. Comput. Methods Appl. Mech. Eng. 62, 145–154 (1987)MATHMathSciNetGoogle Scholar
  39. 39.
    Chen, H., Strikwerda, J.C.: Preconditioning for regular elliptic systems. SIAM J. Numer. Anal. 37(1), 131–151 (1999)MATHMathSciNetGoogle Scholar
  40. 40.
    Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)MATHGoogle Scholar
  41. 41.
    Ciarlet, P: Mathematical Elasticity. Vol. I. Three-dimensional elasticity. Studies in Mathematics and its Applications, vol. 20. North-Holland, Amsterdam (1988)Google Scholar
  42. 42.
    Ciarlet, P.G., Raviart, P.-A.: Maximum principle and uniform convergence for the finite element method. Comput. Methods Appl. Mech. Eng. 2, 17–31 (1973)MATHMathSciNetGoogle Scholar
  43. 43.
    Concus, P., Golub, G.H.: Use of fast direct methods for the efficient numerical solution of nonseparable elliptic equations. SIAM J. Numer. Anal. 10, 1103–1120 (1973)MATHMathSciNetGoogle Scholar
  44. 44.
    Concus, P., Golub, G.H.: A generalized conjugate method for non-symmetric systems of linear equations. In: Glowinski, R., Lions, J.-L. (eds.) Lect. Notes Math. Syst. vol. 134, pp. 56–65. Springer, New York (1976)Google Scholar
  45. 45.
    Czách, L.: The steepest descent method for elliptic differential equations (in Russian). C.Sc. thesis (1955)Google Scholar
  46. 46.
    Dryja, M.: A priori estimates in \(W^{2}_{2}\) in a convex domain for systems of elliptic difference equations (Russian). Ž. Vyčisl. Mat. i Mat. Fiz. 12, 1595–1601, 1632 (1972)MATHMathSciNetGoogle Scholar
  47. 47.
    Dryja, M.: An iterative substructuring method for elliptic mortar finite element problems with discontinuous coefficients. In: Domain Decomposition Methods 10 (Boulder, CO, 1997). Contemp. Math. vol. 218, pp. 94–103. AMS, Providence (1998)Google Scholar
  48. 48.
    D’yakonov, E.G.: On an iterative method for the solution of finite difference equations (in Russian). Dokl. Akad. Nauk SSSR 138, 522–525 (1961)MathSciNetGoogle Scholar
  49. 49.
    D’yakonov, E.G.: The construction of iterative methods based on the use of spectrally equivalent operators. USSR Comput. Math. Math. Phys. 6, 14–46 (1965)Google Scholar
  50. 50.
    Eisenstat, S.C., Elman, H.C., Schultz. M.H.: Variational iterative methods for nonsymmetric systems of linear equations. SIAM J. Numer. Anal. 20(2), 345–357 (1983)MATHMathSciNetGoogle Scholar
  51. 51.
    Elman, H.C., Schultz. M.H.: Preconditioning by fast direct methods for nonself-adjoint nonseparable elliptic equations. SIAM J. Numer. Anal. 23, 44–57 (1986)MATHMathSciNetGoogle Scholar
  52. 52.
    Elman, H.C., Silvester, D.J., Wathen, A.J.: Finite Elements and Fast Iterative Solvers: with Applications in Incompressible Fluid Dynamics, Numerical Mathematics and Scientific Computation. Oxford University Press, New York (2005)Google Scholar
  53. 53.
    Ewing, R.E., Margenov, S.D., Vassilevski, P.S.: Preconditioning the biharmonic equation by multilevel iterations. Math. Balk. (N.S.) 10(1), 121–132 (1996)MATHMathSciNetGoogle Scholar
  54. 54.
    Faber, V., Manteuffel, T.: Necessary and sufficient conditions for the existence of a conjugate gradient method. SIAM J. Numer. Anal. 21(2), 352–362 (1984)MATHMathSciNetGoogle Scholar
  55. 55.
    Faber, V., Manteuffel, T., Parter, S.V.: On the theory of equivalent operators and application to the numerical solution of uniformly elliptic partial differential equations. Adv. Appl. Math. 11, 109–163 (1990)MATHMathSciNetGoogle Scholar
  56. 56.
    Faragó, I., Karátson, J.: Numerical Solution of Nonlinear Elliptic Problems via Preconditioning Operators. Theory and Applications. Advances in Computation, vol. 11. NOVA Science, New York (2002)Google Scholar
  57. 57.
    Funaro, D.: Polynomial Approximation of Differential Equations, Lecture Notes in Physics, New Series, Monographs, vol. 8. Springer, New York (1992)Google Scholar
  58. 58.
    Goldstein, C.I., Manteuffel, T.A., Parter, S.V.: Preconditioning and boundary conditions without H 2 estimates: L 2 condition numbers and the distribution of the singular values. SIAM J. Numer. Anal. 30(2), 343–376 (1993)MATHMathSciNetGoogle Scholar
  59. 59.
    Golub, G.H., O’Leary, D.P.: Some history of the conjugate gradient and Lanczos algorithms: 1948–1976. SIAM Rev. 31(1), 50–102 (1989)MATHMathSciNetGoogle Scholar
  60. 60.
    Golub, G.H., Van Loan, C.F.: Matrix Computations. Johns Hopkins University Press, Baltimore (1983)MATHGoogle Scholar
  61. 61.
    Golub, G.H., Wathen, A.J.: An iteration for indefinite systems and its application to the Navier–Stokes equations. SIAM J. Sci. Comput. 19(2), 530–539 (1998)MATHMathSciNetGoogle Scholar
  62. 62.
    Golub, G.H., Ye, Q.: Inexact preconditioned conjugate gradient method with inner–outer iteration. SIAM J. Sci. Comput. 21(4), 1305–1320 (1999/00)MathSciNetGoogle Scholar
  63. 63.
    Graham, I.G., Hagger, M.J.: Unstructured additive Schwarz-conjugate gradient method for elliptic problems with highly discontinuous coefficients. SIAM J. Sci. Comput. 20, 2041–2066 (1999) (electronic)MATHMathSciNetGoogle Scholar
  64. 64.
    Greenbaum, A.: Diagonal scalings of the Laplacian as preconditioners for other elliptic differential operators. SIAM J. Matrix Anal. Appl. 13, 826–846 (1992)MATHMathSciNetGoogle Scholar
  65. 65.
    Greenbaum, A.: Iterative Methods for Solving Linear Systems. SIAM, Philadelphia (1997)MATHGoogle Scholar
  66. 66.
    Guillard, H., Désidéri, J.-A.: Iterative methods with spectral preconditioning for elliptic equations. Comput. Methods Appl. Mech. Eng. 80(1–3), 305–312 (1990)MATHGoogle Scholar
  67. 67.
    Gunn, J.E.: The numerical solution of \(\nabla\cdot a\nabla u=f\) by a semi-explicit alternating direction iterative method. Numer. Math. 6, 181–184 (1964)MATHMathSciNetGoogle Scholar
  68. 68.
    Gunn, J.E.: The solution of elliptic difference equations by semi-explicit iterative techniques. SIAM J. Numer. Anal. Ser. B 2, 24–45 (1965)MathSciNetCrossRefGoogle Scholar
  69. 69.
    Gustafsson, I.: A class of first order factorization methods. BIT 18(2), 142–156 (1978)MATHMathSciNetGoogle Scholar
  70. 70.
    Hackbusch, W.: Multigrid Methods and Applications, Springer Series in Computational Mathematics, vol. 4. Springer, Berlin (1985)Google Scholar
  71. 71.
    Hestenes, M.R., Stiefel, E.: Methods of conjugate gradients for solving linear systems. J. Res. Natl. Bur. Stand., B 49(6), 409–436 (1952)MATHMathSciNetGoogle Scholar
  72. 72.
    Horgan, C.O.: Korn’s inequalities and their applications in continuum mechanics. SIAM Rev. 37(4), 491–511 (1995)MATHMathSciNetGoogle Scholar
  73. 73.
    Joubert, W., Manteuffel, T.A., Parter, S., Wong, S.-P.: Preconditioning second-order elliptic operators: experiment and theory. SIAM J. Sci. Statist. Comput. 13(1), 259–288 (1992)MATHMathSciNetGoogle Scholar
  74. 74.
    Kantorovich, L.V., Akilov, G.P.: Functional Analysis. Pergamon, Oxford (1984)MATHGoogle Scholar
  75. 75.
    Kaporin, I.E.: New convergence results and preconditioning strategies for the conjugate gradient method. Numer. Linear Algebr. Appl. 1(2), 179–210 (1994)MATHMathSciNetGoogle Scholar
  76. 76.
    Karátson, J.: Mesh independent superlinear convergence estimates of the conjugate gradient method for some equivalent self-adjoint operators. Appl. Math. (Prague) 50(3), 277–290 (2005)MATHGoogle Scholar
  77. 77.
    Karátson, J.: On the superlinear convergence rate of the preconditioned CGM for some nonsymmetric elliptic problems. Numer. Funct. Anal. 28(9–10), 1153–1164 (2007)MATHGoogle Scholar
  78. 78.
    Karátson, J.: Superlinear PCG algorithms: symmetric part preconditioning and boundary conditions. Numer. Funct. Anal. 29, 590–611 (2008)MATHGoogle Scholar
  79. 79.
    Karátson, J., Faragó I.: Variable preconditioning via quasi-Newton methods for nonlinear problems in Hilbert space. SIAM J. Numer. Anal. 41(4), 1242–1262 (2003)MATHMathSciNetGoogle Scholar
  80. 80.
    Karátson, J., Kurics, T.: Superlinearly convergent PCG algorithms for some nonsymmetric elliptic systems. J. Comp. Appl. Math. 212(2), 214–230 (2008)MATHGoogle Scholar
  81. 81.
    Karátson, J., Kurics, T.: Superlinear PCG Methods for FDM Discretizations of Convection-Diffusion Equations. Preprint, ELTE Dept. Appl. Anal. Comp. Math.; 2006-13. Lecture Notes Comp. Sci., Springer (to appear)
  82. 82.
    Karátson, J., Kurics, T., Lirkov, I.: A parallel algorithm for systems of convection–diffusion equations. In: Boyanov, T., et al. (eds.) NMA 2006, Lecture Notes Comp. Sci., vol. 4310, pp. 65–73, Springer, New York (2007)Google Scholar
  83. 83.
    Kim, S.D., Parter, S.V.: Semicirculant preconditioning of elliptic operators. SIAM J. Numer. Anal. 41(2), 767–795 (2003)MATHMathSciNetGoogle Scholar
  84. 84.
    Klawonn, A., Widlund, O.B.: New results on FETI methods for elliptic problems with discontinuous coefficients. In: Numerical analysis 1999 (Dundee), Res. Notes Math., vol. 420, pp. 191–209. Chapman & Hall/CRC, Boca Raton (2000)Google Scholar
  85. 85.
    Knyazev, A, Lashuk, I.: Steepest descent and conjugate gradient methods with variable preconditioning. Electronic. Math. NA/0605767, arXiv. org. (2006–2007)
  86. 86.
    Kuznetsov, Y.A., Rossi, T.: Fast direct method for solving algebraic systems with separable symmetric band matrices. East-West J. Numer. Math. 4(1), 53–68 (1996)MATHMathSciNetGoogle Scholar
  87. 87.
    Kraus, J.: Algebraic multilevel preconditioning of finite element matrices using local Schur complements. Numer. Linear Algebr. Appl. 13, 49–70 (2006)MATHMathSciNetGoogle Scholar
  88. 88.
    Křížek, M., Lin Q.: On diagonal dominance of stiffness matrices in 3D. East-West J. Numer. Math. 3, 59–69 (1995)MATHMathSciNetGoogle Scholar
  89. 89.
    Langer, U., Queck, W.: On the convergence factor of Uzawa’s algorithm. J. Comput. Appl. Math. 15(2), 191–202 (1986)MATHMathSciNetGoogle Scholar
  90. 90.
    Loghin, D.: Green’s functions for preconditioning. DPhil Thesis, Oxford (1999)Google Scholar
  91. 91.
    Manteuffel, T.: The Tchebychev iteration for nonsymmetric linear systems. Numer. Math. 28(3), 307–327 (1977)MATHMathSciNetGoogle Scholar
  92. 92.
    Manteuffel, T., Otto, J.: Optimal equivalent preconditioners. SIAM J. Numer. Anal. 30, 790–812 (1993)MATHMathSciNetGoogle Scholar
  93. 93.
    Manteuffel, T., Parter, S.V.: Preconditioning and boundary conditions. SIAM J. Numer. Anal. 27(3), 656–694 (1990)MATHMathSciNetGoogle Scholar
  94. 94.
    Mayo, A., Greenbaum, A.: Fast parallel iterative solution of Poisson’s and the biharmonic equations on irregular regions. SIAM J. Sci. Statist. Comput. 13(1), 101–118 (1992)MATHMathSciNetGoogle Scholar
  95. 95.
    Mikhlin, S.G.: The Numerical Performance of Variational Methods. Walters–Noordhoff, Alphen aan den Rijn (1971)MATHGoogle Scholar
  96. 96.
    Mikhlin, S.G.: Constants in Some Inequalities of Analysis (translated from the Russian by R. Lehmann). Wiley, Chichester (1986)Google Scholar
  97. 97.
    Nečas, J., Hlaváček, I.: Mathematical Theory of Elastic and Elasto-plastic Bodies: an Introduction, Studies in Applied Mechanics, vol. 3. Elsevier, Amsterdam (1980)Google Scholar
  98. 98.
    Nevanlinna, O.: Convergence of Iterations for Linear Equations. Birkhäuser, Basel (1993)MATHGoogle Scholar
  99. 99.
    Neuberger, J.W.: Sobolev Gradients and Differential Equations, Lecture Notes in Math., No. 1670. Springer, New York (1997)Google Scholar
  100. 100.
    Nitsche, J., Nitsche, J.C.C.: Error estimates for the numerical solution of elliptic differential equations. Arch. Ration. Mech. Anal. 5, 293–306 (1960)MATHMathSciNetGoogle Scholar
  101. 101.
    Repin, S., Sauter, S., Smolianski, A.: A posteriori error estimation for the Poisson equation with mixed Dirichlet/Neumann boundary conditions. J. Comput. Appl. Math. 164–165, 601–612 (2004)MathSciNetGoogle Scholar
  102. 102.
    Rossi, T., Toivanen, J.: Parallel fictitious domain method for a non-linear elliptic Neumann boundary value problem, Czech-US workshop in iterative methods and parallel computing, Part I (Milovy, 1997). Numer. Linear Algebr. Appl. 6(1), 51–60 (1999)MATHMathSciNetGoogle Scholar
  103. 103.
    Rossi, T., Toivanen, J.: A parallel fast direct solver for block tridiagonal systems with separable matrices of arbitrary dimension. SIAM J. Sci. Comput. 20(5), 1778–1796 (1999)MATHMathSciNetGoogle Scholar
  104. 104.
    Saad, Y., Schultz, M.H.: GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Statist. Comput. 7, 856–869 (1986)MATHMathSciNetGoogle Scholar
  105. 105.
    Saad, Y.: Iterative Methods for Sparse Linear Systems. PWS Publ. Co., Boston (1996)MATHGoogle Scholar
  106. 106.
    Saad, Y.: A flexible inner-outer preconditioned GMRES algorithm. SIAM J. Sci. Comput. 14(2), 461–469 (1993)MATHMathSciNetGoogle Scholar
  107. 107.
    Simoncini V., Szyld D.B.: Recent computational developments in Krylov subspace methods for linear systems. Numer. Linear Algebr. Appl. 14(1), 1–59 (2007)MathSciNetMATHGoogle Scholar
  108. 108.
    Simoncini V., Szyld D.B.: Flexible inner-outer Krylov subspace methods. SIAM J. Numer. Anal. 40, 2219–2239 (2003)MATHMathSciNetGoogle Scholar
  109. 109.
    Sundqvist, P.: Numerical computations with fundamental solutions. PhD thesis, Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology, Uppsala University (2005)Google Scholar
  110. 110.
    Swarztrauber, P.N.: A direct method for the discrete solution of separable elliptic equations. SIAM J. Numer. Anal. 11, 1136–1150 (1974)MATHMathSciNetGoogle Scholar
  111. 111.
    Swarztrauber, P.N.: The methods of cyclic reduction, Fourier analysis and the FACR algorithm for the discrete solution of Poisson’s equation on a rectangle. SIAM Rev. 19(3), 490–501 (1977)MATHMathSciNetGoogle Scholar
  112. 112.
    Temam, R.: Navier-Stokes Equations. Theory and Numerical Analysis, Studies in Mathematics and its Applications, vol. 2. North-Holland, Amsterdam (1977)Google Scholar
  113. 113.
    Vassilevski, P.S.: Fast algorithm for solving a linear algebraic problem with separable variables. C. R. Acad. Bulgare Sci. 37(3), 305–308 (1984)MATHMathSciNetGoogle Scholar
  114. 114.
    Vladimirov, V.S.: Equations of Mathematical Physics (translated from the Russian by E. Yankovsky). Mir, Moscow (1984)Google Scholar
  115. 115.
    Winter, R.: Some superlinear convergence results for the conjugate gradient method. SIAM J. Numer. Anal. 17, 14–17 (1980)MathSciNetGoogle Scholar
  116. 116.
    Widlund, O.: On the use of fast methods for separable finite difference equations for the solution of general elliptic problems. In: Rose, D.J., Willoughby, R.A. (eds.) Sparse Matrices and Applications, pp. 121–134. Plenum, New York (1972)Google Scholar
  117. 117.
    Widlund, O.: A Lanczos method for a class of non-symmetric systems of linear equations. SIAM J. Numer. Anal. 15, 801–812 (1978)MATHMathSciNetGoogle Scholar
  118. 118.
    Young, D.M.: Iterative Solution of Large Linear Systems. Academic, New York (1971)MATHGoogle Scholar
  119. 119.
    Zlatev, Z.: Computer Treatment of Large Air Pollution Models. Kluwer Academic, Dordrecht (1995)Google Scholar

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© Springer Science+Business Media, LLC. 2008

Authors and Affiliations

  1. 1.Department of Information TechnologyUppsala UniversityUppsalaSweden
  2. 2.Institute of Geonics AS CROstravaCzech Republic
  3. 3.Department of Applied AnalysisELTE UniversityBudapestHungary

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