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Stopping criteria for iterations in finite element methods

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Summary.

This work extends the results of Arioli [1], [2] on stopping criteria for iterative solution methods for linear finite element problems to the case of nonsymmetric positive-definite problems. We show that the residual measured in the norm induced by the symmetric part of the inverse of the system matrix is relevant to convergence in a finite element context. We then use Krylov solvers to provide alternative ways of calculating or estimating this quantity and present numerical experiments which validate our criteria.

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References

  1. Arioli, M.: A stopping criterion for the conjugate gradient algorithm in a finite element method framework. Numer. Math. 97(1), 1–24 (2004)

    Google Scholar 

  2. Arioli, M., Noulard, E., Russo, A.: Stopping criteria for iterative methods: applications to PDEs. Calcolo 38, 97–112 (2001)

    Article  Google Scholar 

  3. Babuška, I.: Error bounds for finite element methods. Numer. Math. 16, 322–333 (1971)

    Google Scholar 

  4. Brezzi, F., Bathe, K.J.: A discourse on the stability conditions for mixed finite element formulations. Comp. Meth. Appl. Mech. Engrg. 82, 27–57 (1990)

    Article  MATH  Google Scholar 

  5. Ciarlet, P.G.: The finite element method for elliptic problems. North Holland, Amsterdam, 1978

  6. Concus, P., Golub, G.H.: A generalized conjugate gradient method for nonsymmetric systems of linear equations. In: R. Glowinski, J.L. Lions, (eds.), Proc. Second Internat. Symp. on Computing Methods in Applied Sciences and Engineering, volume 134 of Lecture Notes in Economics and Mathematical Systems, Berlin, 1976. Springer Verlag

  7. Deuflhard, P.: Cascadic conjugate gradient methods for elliptic partial differential equations. Algorithm and numerical results. In: J. Xu, D. Keyes, (eds.), Proceedings of the 7th International Conference on Domain Decomposition Methods, AMS Providence, 1993, pp. 29–42

  8. Deuflhard, P.: Cascadic conjugate gradient methods for elliptic partial differential equations: algorithm and numerical results. Contemporary Mathematics 180, 29–42 (1994)

    MATH  Google Scholar 

  9. Deuflhard, P., Bornemann, F.A.: The cascadic multigrid method for elliptic problems. Numerische Mathematik 75, 135–152 (1996)

    Article  MATH  Google Scholar 

  10. Golub, G.H., Meurant, G.: Matrices, moments and quadrature II; how to compute the norm of the error in iterative methods. BIT 37(3), 687–705 (1997)

    MATH  Google Scholar 

  11. Golub, G.H., Strakoš, Z.: Estimates in quadratic formulas. Numer. Algorithms 8, 2–4 (1994)

    MATH  Google Scholar 

  12. Higham, N.J.: Accuracy and stability of numerical algorithms. SIAM, 2002

  13. Horn, R.A., Johnson, C.R.: Matrix analysis. Cambridge University Press, 1985

  14. Horn, R.A., Johnson, C.R.: Topics in matrix analysis. Cambridge University Press, 1991

  15. Loghin, D., Wathen, A.J.: Analysis of block preconditioners for saddle-point problems. Technical Report 13, Oxford University Computing Laboratory, 2002, Submitted to SISC

  16. Meurant, G.: Numerical experiments in computing bounds for the norm of the error in the preconditioned conjugate gradient algorithm. Numer. Algoithms 22, 353–365 (1999)

    Article  MATH  Google Scholar 

  17. Rigal, J.L., Gaches, J.: On the compatibility of a given solution with the data of a linear system. J. Assoc. Comput. Mach. 14(3), 543–548 (1967)

    MATH  Google Scholar 

  18. Starke, G.: Field-of-values analysis of preconditioned iterative methods for non-symmetric elliptic problems. Numer. Math. 78, 103–117 (1997)

    Article  MATH  Google Scholar 

  19. Strakoš, Z., Tichý, P.: On error estimation by conjugate gradient method and why it works in finite precision computations. Electronic Transactions on Numerical Analysis 13, 56–80 (2002)

    Google Scholar 

  20. Strang, W.G., Fix, G.J.: An analysis of the finite element method. Prentice-Hall, 1973

  21. Widlund, O.: A Lanczos method for a class of nonsymmetric systems of linear equations. SIAM J. Numer. Anal. 15(4), 1978

    MATH  Google Scholar 

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Authors and Affiliations

  1. Atlas Centre, Rutherford Appleton Laboratory, Oxon, OX11 0QX, UK

    M. Arioli

  2. CERFACS, 42 ave G. Coriolis, Toulouse, 31057, France

    D. Loghin

  3. Oxford University Computing Laboratory, Parks Road, Oxford, OX1 3QD, UK

    A. J. Wathen

Authors
  1. M. Arioli
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  2. D. Loghin
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  3. A. J. Wathen
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Correspondence to M. Arioli.

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Mathematics Subject Classification (2000): 65N30, 65F10, 65F35

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Arioli, M., Loghin, D. & Wathen, A. Stopping criteria for iterations in finite element methods. Numer. Math. 99, 381–410 (2005). https://doi.org/10.1007/s00211-004-0568-z

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  • DOI: https://doi.org/10.1007/s00211-004-0568-z

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