A comparison of some efficient numerical methods for a nonlinear elliptic problem

Balázs Kovács

Open Mathematics (2012)

  • Volume: 10, Issue: 1, page 217-230
  • ISSN: 2391-5455

Abstract

top
The aim of this paper is to compare and realize three efficient iterative methods, which have mesh independent convergence, and to propose some improvements for them. We look for the numerical solution of a nonlinear model problem using FEM discretization with gradient and Newton type methods. Three numerical methods have been carried out, namely, the gradient, Newton and quasi-Newton methods. We have solved the model problem with these methods, we have investigated the differences between them and analyzed their behavior, efficiency and mesh independence. We also compare the theoretical results to the numerical ones, and finally we propose some improvements which we also investigate.

How to cite

top

Balázs Kovács. "A comparison of some efficient numerical methods for a nonlinear elliptic problem." Open Mathematics 10.1 (2012): 217-230. <http://eudml.org/doc/269774>.

@article{BalázsKovács2012,
abstract = {The aim of this paper is to compare and realize three efficient iterative methods, which have mesh independent convergence, and to propose some improvements for them. We look for the numerical solution of a nonlinear model problem using FEM discretization with gradient and Newton type methods. Three numerical methods have been carried out, namely, the gradient, Newton and quasi-Newton methods. We have solved the model problem with these methods, we have investigated the differences between them and analyzed their behavior, efficiency and mesh independence. We also compare the theoretical results to the numerical ones, and finally we propose some improvements which we also investigate.},
author = {Balázs Kovács},
journal = {Open Mathematics},
keywords = {Radiative cooling problem; Iterative methods; Stepwise variable preconditioning; Numerical experiments; radiative cooling problem; iterative methods; stepwise variable preconditioning; numerical experiments; nonlinear elliptic problem; gradient method; Newton method; quasi-Newton method},
language = {eng},
number = {1},
pages = {217-230},
title = {A comparison of some efficient numerical methods for a nonlinear elliptic problem},
url = {http://eudml.org/doc/269774},
volume = {10},
year = {2012},
}

TY - JOUR
AU - Balázs Kovács
TI - A comparison of some efficient numerical methods for a nonlinear elliptic problem
JO - Open Mathematics
PY - 2012
VL - 10
IS - 1
SP - 217
EP - 230
AB - The aim of this paper is to compare and realize three efficient iterative methods, which have mesh independent convergence, and to propose some improvements for them. We look for the numerical solution of a nonlinear model problem using FEM discretization with gradient and Newton type methods. Three numerical methods have been carried out, namely, the gradient, Newton and quasi-Newton methods. We have solved the model problem with these methods, we have investigated the differences between them and analyzed their behavior, efficiency and mesh independence. We also compare the theoretical results to the numerical ones, and finally we propose some improvements which we also investigate.
LA - eng
KW - Radiative cooling problem; Iterative methods; Stepwise variable preconditioning; Numerical experiments; radiative cooling problem; iterative methods; stepwise variable preconditioning; numerical experiments; nonlinear elliptic problem; gradient method; Newton method; quasi-Newton method
UR - http://eudml.org/doc/269774
ER -

References

top
  1. [1] Brezinski C., A classification of quasi-Newton methods, Numer. Algorithms, 2003, 33(1–4), 123–135 http://dx.doi.org/10.1023/A:1025551602679 Zbl1030.65053
  2. [2] Dennis J.E., Jr., Moré J.J., Quasi-Newton methods, motivation and theory, SIAM Rev., 1977, 19(1), 46–89 http://dx.doi.org/10.1137/1019005 Zbl0356.65041
  3. [3] Faragó I., Karátson J., Numerical Solution of Nonlinear Elliptic Problems via Preconditioning Operators: Theory and Applications, Adv. Comput. Theory Pract., 11, Nova Science Publishers, Hauppauge, 2002 Zbl1030.65117
  4. [4] Karátson J., Faragó I., Variable preconditioning via quasi-Newton methods for nonlinear problems in Hilbert space, SIAM J. Numer. Anal., 2003, 41(4), 1242–1262 http://dx.doi.org/10.1137/S0036142901384277 Zbl1130.65309
  5. [5] Keller H.B., Elliptic boundary value problems suggested by nonlinear diffusion processes, Arch. Rational Mech. Anal., 1969, 35(5), 363–381 http://dx.doi.org/10.1007/BF00247683 Zbl0188.17102
  6. [6] Liao S., Su J., Chwang A.T., Series solutions for a nonlinear model of combined convective and radiative cooling of a spherical body, International Journal of Heat and Mass Transfer, 2006, 49(15–16), 2437–2445 http://dx.doi.org/10.1016/j.ijheatmasstransfer.2006.01.030 Zbl1189.76549
  7. [7] Schlichenmaier R., Bruls J.H.M.J., Schüssler M., Radiative cooling of a hot flux tube in the solar photosphere, Astronomy and Astrophysics, 1999, 349, 961–973 
  8. [8] Sen M., Analytical Heat Transfer, lecture notes available at http://nd.edu/_msen/Teaching/IntHT/IntHTNotes.pdf 
  9. [9] Vladimirov V.S., Equations of Mathematical Physics, Mir, Moscow, 1984 
  10. [10] Zeidler E., Nonlinear Functional Analysis and its Applications. III, Springer, New York, 1985 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.