Multilevel correction adaptive finite element method for semilinear elliptic equation

Qun Lin; Hehu Xie; Fei Xu

Applications of Mathematics (2015)

  • Volume: 60, Issue: 5, page 527-550
  • ISSN: 0862-7940

Abstract

top
A type of adaptive finite element method is presented for semilinear elliptic problems based on multilevel correction scheme. The main idea of the method is to transform the semilinear elliptic equation into a sequence of linearized boundary value problems on the adaptive partitions and some semilinear elliptic problems on very low dimensional finite element spaces. Hence, solving the semilinear elliptic problem can reach almost the same efficiency as the adaptive method for the associated boundary value problem. The convergence and optimal complexity of the new scheme can be derived theoretically and demonstrated numerically.

How to cite

top

Lin, Qun, Xie, Hehu, and Xu, Fei. "Multilevel correction adaptive finite element method for semilinear elliptic equation." Applications of Mathematics 60.5 (2015): 527-550. <http://eudml.org/doc/271565>.

@article{Lin2015,
abstract = {A type of adaptive finite element method is presented for semilinear elliptic problems based on multilevel correction scheme. The main idea of the method is to transform the semilinear elliptic equation into a sequence of linearized boundary value problems on the adaptive partitions and some semilinear elliptic problems on very low dimensional finite element spaces. Hence, solving the semilinear elliptic problem can reach almost the same efficiency as the adaptive method for the associated boundary value problem. The convergence and optimal complexity of the new scheme can be derived theoretically and demonstrated numerically.},
author = {Lin, Qun, Xie, Hehu, Xu, Fei},
journal = {Applications of Mathematics},
keywords = {semilinear elliptic problem; multilevel correction; adaptive finite element method; semilinear elliptic problem; multilevel correction; adaptive finite element method},
language = {eng},
number = {5},
pages = {527-550},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Multilevel correction adaptive finite element method for semilinear elliptic equation},
url = {http://eudml.org/doc/271565},
volume = {60},
year = {2015},
}

TY - JOUR
AU - Lin, Qun
AU - Xie, Hehu
AU - Xu, Fei
TI - Multilevel correction adaptive finite element method for semilinear elliptic equation
JO - Applications of Mathematics
PY - 2015
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 60
IS - 5
SP - 527
EP - 550
AB - A type of adaptive finite element method is presented for semilinear elliptic problems based on multilevel correction scheme. The main idea of the method is to transform the semilinear elliptic equation into a sequence of linearized boundary value problems on the adaptive partitions and some semilinear elliptic problems on very low dimensional finite element spaces. Hence, solving the semilinear elliptic problem can reach almost the same efficiency as the adaptive method for the associated boundary value problem. The convergence and optimal complexity of the new scheme can be derived theoretically and demonstrated numerically.
LA - eng
KW - semilinear elliptic problem; multilevel correction; adaptive finite element method; semilinear elliptic problem; multilevel correction; adaptive finite element method
UR - http://eudml.org/doc/271565
ER -

References

top
  1. Adams, R. A., Sobolev Spaces, Pure and Applied Mathematics 65. A Series of Monographs and Textbooks Academic Press, New York (1975). (1975) Zbl0314.46030MR0450957
  2. Babuška, I., Miller, A., 10.1016/0045-7825(87)90114-9, Comput. Methods Appl. Mech. Eng. 61 (1987), 1-40. (1987) Zbl0593.65064MR0880421DOI10.1016/0045-7825(87)90114-9
  3. Babuška, I., Rheinboldt, W. C., 10.1137/0715049, SIAM J. Numer. Anal. 15 (1978), 736-754. (1978) Zbl0398.65069MR0483395DOI10.1137/0715049
  4. Babuška, I., Strouboulis, T., The Finite Element Method and Its Reliability, Numerical Mathematics and Scientific Computation Clarendon Press, Oxford (2001). (2001) MR1857191
  5. Babuška, I., Vogelius, M., 10.1007/BF01389757, Numer. Math. 44 (1984), 75-102. (1984) Zbl0574.65098MR0745088DOI10.1007/BF01389757
  6. Brenner, S. C., Scott, L. R., 10.1007/978-1-4757-4338-8_7, Texts in Applied Mathematics 15 Springer, New York (1994). (1994) Zbl0804.65101MR1278258DOI10.1007/978-1-4757-4338-8_7
  7. Cascon, J. M., Kreuzer, C., Nochetto, R. H., Siebert, K. G., 10.1137/07069047X, SIAM J. Numer. Anal. 46 (2008), 2524-2550. (2008) Zbl1176.65122MR2421046DOI10.1137/07069047X
  8. Ciarlet, P. G., The Finite Element Method for Elliptic Problems, Studies in Mathematics and Its Applications. Vol. 4 North-Holland Publishing Company, Amsterdam (1978). (1978) Zbl0383.65058MR0520174
  9. Dörfler, W., 10.1137/0733054, SIAM J. Numer. Anal. 33 (1996), 1106-1124. (1996) Zbl0854.65090MR1393904DOI10.1137/0733054
  10. He, L., Zhou, A., Convergence and optimal complexity of adaptive finite element methods for elliptic partial differential equations, Int. J. Numer. Anal. Model. 8 (2011), 615-640. (2011) MR2805661
  11. Holst, M., McCammom, J. A., Yu, Z., Zhou, Y., Zhu, Y., 10.4208/cicp.081009.130611a, Commun. Comput. Phys. 11 (2012), 179-214. (2012) MR2841952DOI10.4208/cicp.081009.130611a
  12. Lin, Q., Xie, H., A multilevel correction type of adaptive finite element method for Steklov eigenvalue problems, Proc. Internat. Conference `Applications of Mathematics', Prague, 2012. In Honor of the 60th Birthday of M. Křížek Academy of Sciences of the Czech Republic, Institute of Mathematics, Prague (2012), 134-143 J. Brandts et al. (2012) Zbl1313.65298MR3204407
  13. Lin, Q., Xie, H., 10.1090/S0025-5718-2014-02825-1, Math. Comput. 84 (2015), 71-88. (2015) Zbl1307.65159MR3266953DOI10.1090/S0025-5718-2014-02825-1
  14. Mekchay, K., Nochetto, R. H., 10.1137/04060929X, SIAM J. Numer. Anal. 43 (2005), 1803-1827. (2005) Zbl1104.65103MR2192319DOI10.1137/04060929X
  15. Morin, P., Nochetto, R. H., Siebert, K. G., 10.1137/S0036142999360044, SIAM J. Numer. Anal. 38 (2000), 466-488. (2000) Zbl0970.65113MR1770058DOI10.1137/S0036142999360044
  16. Morin, P., Nochetto, R. H., Siebert, K. G., 10.1137/S0036144502409093, SIAM Rev. 44 (2002), 631-658. (2002) Zbl1016.65074MR1980447DOI10.1137/S0036144502409093
  17. Stevenson, R., 10.1007/s10208-005-0183-0, Found. Comput. Math. 7 (2007), 245-269. (2007) Zbl1136.65109MR2324418DOI10.1007/s10208-005-0183-0
  18. Stevenson, R., 10.1090/S0025-5718-07-01959-X, Math. Comput. 77 (2008), 227-241. (2008) Zbl1131.65095MR2353951DOI10.1090/S0025-5718-07-01959-X
  19. Xie, H., A multilevel correction type of adaptive finite element method for eigenvalue problems, ArXiv:1201.2308 (2012). (2012) MR3204407

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.