Skip to main content
×
Home

A MULTIPLICATIVE SCHWARZ ALGORITHM FOR THE NONLINEAR COMPLEMENTARITY PROBLEM WITH AN M-FUNCTION

  • YINGJUN JIANG (a1) and JINPING ZENG (a2)
Abstract
Abstract

A multiplicative Schwarz iteration algorithm is presented for solving the finite-dimensional nonlinear complementarity problem with an M-function. The monotone convergence of the iteration algorithm is obtained with special choices of initial values. Moreover, by applying the concept of weak regular splitting, the weighted max-norm bound is derived for the iteration errors.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      A MULTIPLICATIVE SCHWARZ ALGORITHM FOR THE NONLINEAR COMPLEMENTARITY PROBLEM WITH AN M-FUNCTION
      Available formats
      ×
      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about sending content to Dropbox.

      A MULTIPLICATIVE SCHWARZ ALGORITHM FOR THE NONLINEAR COMPLEMENTARITY PROBLEM WITH AN M-FUNCTION
      Available formats
      ×
      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about sending content to Google Drive.

      A MULTIPLICATIVE SCHWARZ ALGORITHM FOR THE NONLINEAR COMPLEMENTARITY PROBLEM WITH AN M-FUNCTION
      Available formats
      ×
Copyright
Corresponding author
For correspondence; e-mail: zengjp@dgut.edu.cn
Footnotes
Hide All

This work is supported by the National Natural Science Foundation of China (Grants 10901027, 10971058).

Footnotes
References
Hide All
[1]Bedea L., ‘On the Schwarz alternating method with more than two subdomains for nonlinear monotone problems’, SIAM J. Numer. Anal. 28 (1991), 179204.
[2]Benzi M., Frommer A. and Nabben R., ‘Algebraic theory of multiplicative Schwarz methods’, Numer. Math. 89 (2001), 605639.
[3]Berman A. and Plemmons R. J., Nonnegative Matrices in the Mathematical Sciences (Academic Press, New York, 1979).
[4]Hoffmann K.-H. and Zou J., ‘Parallel solution of variational inequality problems with nonlinear source terms’, IMA J. Numer. Anal. 16 (1996), 3145.
[5]Hoppe R. H. W., ‘Multigrid algorithms for variational inequalities’, SIAM J. Numer. Anal. 24 (1987), 10461065.
[6]Householder A. S., The Theory of Matrices in Numerical Analysis (Blaisdell, Waltham, MA, 1964).
[7]Isac G., Complementarity Problems (Springer, Berlin, 1992).
[8]Jiang Y. J. and Zeng J. P., ‘An L -error estimate for finite element solution of nonlinear elliptic problem with a source term’, Appl. Math. Comput. 174 (2006), 134149.
[9]Kuznetsov Y., Neittaanmäki P. and Tarvainen P., ‘Overlapping domain decomposition methods for the obstacle problem’, in: Domain Decomposition Methods in Science and Engineering, (eds. Kuznetsov Y., Peraux J., Quarteroni A. and Widlund O.) (American Mathematical Society, Providence, RI, 1994), pp. 271277.
[10]Kuznetsov Y., Neittaanmäki P. and Tarvainen P., ‘Schwarz methods for obstacle problems with convection diffusion operators’, in: Domain Decomposition Methods in Scientific and Engineering Computing, Contemporary Mathematics, 180 (eds. Keyes D. E. and Xu J. C.) (American Mathematical Society, Providence, RI, 1995), pp. 251256.
[11]Lions P. L., ‘On the Schwarz alternating method. I’. First International Symposium on Domain Decomposition Methods for Partial Differential Equations (Paris, 1987) (eds. Glowinshi R., Golub G. H., Meurant G. A. and Périaux J.) (SIAM, Philadelphia, PA, 1988), pp. 142.
[12]Lions P. L., ‘On the Schwarz alternating method. II’. Stochastic Interpretation and Order Properties, Proceedings of Domain Decomposition Methods (Los Angeles, CA, 1988) (eds. Chan T. F., Glowinski R., Périaux J. and Widlund O. B.) (SIAM, Philadelphia, PA, 1989), pp. 4770.
[13] T., Shih T. M. and Liem C. B., ‘Parallel algorithms for variational inequalities based on domain decomposition’, J. Comput. Math. 9 (1991), 340349.
[14] T., Shih T. M. and Liem C. B., Domain Decomposition Methods — New Numerical Technique for Solving PDE (Science Press, Beijing, 1992).
[15]Luo Z. Q. and Tseng P., ‘Error bound and convergence analysis of matrix splitting algorithms for the affine variational inequality problem’, SIAM J. Optim. 2 (1992), 4354.
[16]Machida N., Fukushima M. and Ibaraki T., ‘A multisplitting method for symmetric linear complementarity problems’, J. Comput. Appl. Math. 62 (1995), 217227.
[17]Moré J. and Rheinboldt W. C., ‘On P- and S-functions and related class of n-dimensional nonlinear mappings’, Linear Algebra Appl. 6 (1973), 4568.
[18]Ortega J. M. and Rheinboldt W. C., Iterative Solution of Nonlinear Equations in Several Variables (Academic Press, New York, 1970).
[19]Scarpini F., ‘The alternative Schwarz method applied to some biharmonic variational inequalities’, Calcolo 27 (1990), 5772.
[20]Smith B., Bjørstad P. and Gropp W., Domain Decomposition (Cambridge University Press, Cambridge, 1996).
[21]Tarvainen P., ‘Block relaxation methods for algebraic obstacle problems with M-matrices: theory and applications’, Doctoral Thesis, Department of Mathematics, University of Jyväskylä, 1994.
[22]Tarvainen P., ‘On the convergence of block relaxation methods for algebraic obstacle problems with M-matrices’, East-West J. Numer. Math. 4 (1996), 6982.
[23]Varga R., Matrix Iterative Analysis (Prentice Hall, Englewood Cliffs, NJ, 1962).
[24]Zeng J. P., Li D. H. and Fukushima M., ‘Weighted max-norm estimate of additive Schwarz iteration algorithm for solving linear complementarity problems’, J. Comput. Appl. Math. 131 (2001), 114.
[25]Zeng J. P. and Zhou S. Z., ‘On monotone and geometric convergence of Schwarz methods for two-sided obstacle problems’, SIAM J. Numer. Anal. 35 (1998), 600616.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Keywords:

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 24 *
Loading metrics...

Abstract views

Total abstract views: 26 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 24th November 2017. This data will be updated every 24 hours.