A variational parameters-to-estimate-free nonlinear solver

  • Svetozara I. Petrova
  • Panayot S. Vassilevski
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1196)


The paper introduces and analyzes an extension of the simple steepest descent variational iterative method for solving nonlinear equations defined from non-differentiable nonlinear mappings. This method is originally proposed for solving a class of nonlinear equations that arise in inner-outer iterative methods for solving linear equations with matrices of a two-by-two block form, see Axelsson and Vassilevski [2]. With minor modifications it turns out to be applicable to a more general class of nonlinear equations defined from non-differentiable mappings that are however sufficiently close to differentiable mappings in a neighborhood of the solution. An extension of the preconditioned steepest descent variational (PSD) method for solving semi-linear elliptic PDEs is illustrated with numerical experiments for a class of finite element discretization techniques based on two subspaces. The convergence of the PSD versus versions of Newton's method is compared and discussed.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    E.L. Allgower, K. Böhmer, F.A. Potra and W.C. Rheinboldt, A mesh-independence principle for operator equations and their discretizations, SIAM J. Numer. Anal., 23, 1986, 160–169.Google Scholar
  2. 2.
    O. Axelsson and P.S. Vassilevski, Construction of variable-step preconditioners for inner-outer iteration methods, Iterative Methods in Linear Algebra (eds. R. Beauwens and P. de Groen), Elsevier Science Publishers B.V. (North-Holland), Amsterdam, 1992, 1–14.Google Scholar
  3. 3.
    O. Axelsson and I.E. Kaporin, On the solution of nonlinear equations for non-differentiable mappings, In fast solvers for flow problems (W. Hackbusch, ed.), Vieweg-Verlag, Braunschweig/Wiesbaden, 1994, 38–51.Google Scholar
  4. 4.
    O. Axelsson, On mesh independence and Newton-type methods, Applications of Mathematics, 38, 1993, 249–265.Google Scholar
  5. 5.
    R.E. Bank and D.J. Rose, Global approximate Newton methods, Numer. Math., 37, 1981, 279–295.Google Scholar
  6. 6.
    M.O. Bristeau, R. Glowinski, J. Periaux, P. Perrier, and O. Pironneau, On the numerical solution of nonlinear problems in fluid dynamics by least squares and finite element methods (I) Least squares formulations and conjugate gradient solution of the continuous problems, Comp. Meth. Appl. Mech. Eng., 17/18, 1979, 619–657.Google Scholar
  7. 7.
    P.N. Brown and Y. Saad, Hybrid Krylov methods for nonlinear systems of equations, SIAM J. Sci. Stat. Comput., 11, 1990, 450–481.Google Scholar
  8. 8.
    R.S. Dembo, S.C. Eisenstat and T. Steihaug, Inexact Newton methods, SIAM J. Numer. Anal., 19, 1982, 400–408.Google Scholar
  9. 9.
    V. Giarault and P.-A. Raviart, Finite Element Methods for Navier-Stokes equations, Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, 1986.Google Scholar
  10. 10.
    R. Glowinski, H.B. Heller and L. Reinhart, Continuation-conjugate gradient methods for the least squares solution of nonlinear boundary value problems, SIAM J. Sci. Stat. Comput., 6, 1985, 793–832.Google Scholar
  11. 11.
    R. Glowinski, J. Periaux, and Q.V. Dihn, Domain decomposition methods for non-linear problems in fluid dynamics, Rapports de Recherche, N∘ 147, 1982, INRIA, Domaine de Voluceau, Rocquencourt, France.Google Scholar
  12. 12.
    W. Hackbusch, Multigrid Methods and Applications, Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, 1985.Google Scholar
  13. 13.
    L.V. Kantorovich and G.P. Akilov, Functional Analysis (Russian), Nauka, Moscow, 1977.Google Scholar
  14. 14.
    I.E. Kaporin and O. Axelsson, On a class of nonlinear equation solvers based on the residual norm reduction over a sequence of affine subspaces, SIAM J. Sci. Comput., 16, 1994, 228–249.Google Scholar
  15. 15.
    A.A. Reusken, Convergence of the multilevel full approximation scheme (FAS) including the V-cycle, Preprint # 492, Department of Mathematics, University of Utrecht, The Netherlands, 1987.Google Scholar
  16. 16.
    G. Vainikko, Galerkin's perturbation method and the general theory of approximation for nonlinear equations, USSR Comp. Math. Phys., 7, 1967, 1–41.Google Scholar
  17. 17.
    J. Xu, Two-grid finite element discretization for nonlinear elliptic equations, Report No. AM 105, Department of Mathematics, Penn State University, 1992, (SINUM, to appear).Google Scholar
  18. 18.
    T.J. Ypma, Local convergence of inexact Newton methods, SIAM J. Numer. Anal., 21, 1984, 583–590.Google Scholar
  19. 19.
    D. Zinćello, A class of approximate methods for solving operator equations with nondifferentiable operators, Dopovidi Akad. Nauk Ukrain SSR, 1963, 852–856.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Svetozara I. Petrova
    • 1
  • Panayot S. Vassilevski
    • 1
  1. 1.Central Laboratory of Parallel ProcessingBulgarian Academy of SciencesSofiaBulgaria

Personalised recommendations