Verification of functional a posteriori error estimates for obstacle problem in 1D

Petr Harasim; Jan Valdman

Kybernetika (2013)

  • Volume: 49, Issue: 5, page 738-754
  • ISSN: 0023-5954

Abstract

top
We verify functional a posteriori error estimate for obstacle problem proposed by Repin. Simplification into 1D allows for the construction of a nonlinear benchmark for which an exact solution of the obstacle problem can be derived. Quality of a numerical approximation obtained by the finite element method is compared with the exact solution and the error of approximation is bounded from above by a majorant error estimate. The sharpness of the majorant error estimate is discussed.

How to cite

top

Harasim, Petr, and Valdman, Jan. "Verification of functional a posteriori error estimates for obstacle problem in 1D." Kybernetika 49.5 (2013): 738-754. <http://eudml.org/doc/260686>.

@article{Harasim2013,
abstract = {We verify functional a posteriori error estimate for obstacle problem proposed by Repin. Simplification into 1D allows for the construction of a nonlinear benchmark for which an exact solution of the obstacle problem can be derived. Quality of a numerical approximation obtained by the finite element method is compared with the exact solution and the error of approximation is bounded from above by a majorant error estimate. The sharpness of the majorant error estimate is discussed.},
author = {Harasim, Petr, Valdman, Jan},
journal = {Kybernetika},
keywords = {obstacle problem; a posteriori error estimate; functional majorant; finite element method; variational inequalities; Uzawa algorithm; obstacle problem; a-posteriori error estimate; functional majorant; finite element method; variational inequalities; Uzawa algorithm},
language = {eng},
number = {5},
pages = {738-754},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Verification of functional a posteriori error estimates for obstacle problem in 1D},
url = {http://eudml.org/doc/260686},
volume = {49},
year = {2013},
}

TY - JOUR
AU - Harasim, Petr
AU - Valdman, Jan
TI - Verification of functional a posteriori error estimates for obstacle problem in 1D
JO - Kybernetika
PY - 2013
PB - Institute of Information Theory and Automation AS CR
VL - 49
IS - 5
SP - 738
EP - 754
AB - We verify functional a posteriori error estimate for obstacle problem proposed by Repin. Simplification into 1D allows for the construction of a nonlinear benchmark for which an exact solution of the obstacle problem can be derived. Quality of a numerical approximation obtained by the finite element method is compared with the exact solution and the error of approximation is bounded from above by a majorant error estimate. The sharpness of the majorant error estimate is discussed.
LA - eng
KW - obstacle problem; a posteriori error estimate; functional majorant; finite element method; variational inequalities; Uzawa algorithm; obstacle problem; a-posteriori error estimate; functional majorant; finite element method; variational inequalities; Uzawa algorithm
UR - http://eudml.org/doc/260686
ER -

References

top
  1. Ainsworth, M., Oden, J. T., A Posteriori Error Estimation in Finite Element Analysis., Wiley and Sons, New York 2000. Zbl1008.65076MR1885308
  2. Babuška, I., Strouboulis, T., The Finite Element Method and its Reliability., Oxford University Press, New York 2001. MR1857191
  3. Bangerth, W., Rannacher, R., Adaptive Finite Element Methods for Differential Equations., Birkhäuser, Berlin 2003. Zbl1020.65058MR1960405
  4. Braess, D., Hoppe, R. H. W., Schöberl, J., 10.1007/s00791-008-0104-2, Comp. Visual. Sci. 11 (2008), 351-362. MR2425501DOI10.1007/s00791-008-0104-2
  5. Brezi, F., Hager, W. W., Raviart, P. A., 10.1007/BF01404345, Numer. Math. 28 (1977), 431-443. MR0448949DOI10.1007/BF01404345
  6. Buss, H., Repin, S., A posteriori error estimates for boundary value problems with obstacles., In: Proc. 3rd European Conference on Numerical Mathematics and Advanced Applications, Jÿvaskylä 1999, World Scientific 2000, pp. 162-170. Zbl0968.65041MR1936177
  7. Carstensen, C., Merdon, C., 10.1002/num.21728, Numer. Methods Partial Differential Equations 29 (2013), 667-�692. MR3022903DOI10.1002/num.21728
  8. Dostál, Z., Optimal Quadratic Programming Algorithms., Springer 2009. MR2492434
  9. Falk, R. S., 10.1090/S0025-5718-1974-0391502-8, Math. Comput. 28 (1974), 963-971. Zbl0297.65061MR0391502DOI10.1090/S0025-5718-1974-0391502-8
  10. Fuchs, M., Repin, S., 10.1080/01630563.2011.571802, Numer. Funct. Anal. Optim. 32 (2011), 610-640. MR2795532DOI10.1080/01630563.2011.571802
  11. Glowinski, R., Lions, J. L., Trémolieres, R., Numerical Analysis of Variational Inequalities., North-Holland 1981. Zbl0463.65046MR0635927
  12. Hlaváček, I., Haslinger, J., Nečas, J., Lovíšek, J., Solution of Variational Inequalities in Mechanics., Applied Mathematical Sciences 66, Springer-Verlag, New York 1988. Zbl0654.73019MR0952855
  13. Kikuchi, N., Oden, J. T., Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods., SIAM 1995. Zbl0685.73002MR0961258
  14. Kraus, J., Tomar, S., 10.1002/nme.3103, Internat. J. Numer. Meth. Engrg. 86 (2011), 1175-1196. Zbl1235.65130MR2817075DOI10.1002/nme.3103
  15. Lions, J. L., Stampacchia, G., 10.1002/cpa.3160200302, Comm. Pure Appl. Math. XX(3) (1967), 493-519. Zbl0152.34601MR0216344DOI10.1002/cpa.3160200302
  16. Neittaanmäki, P., Repin, S., Reliable Methods for Computer Simulation (Error Control and a Posteriori Estimates)., Elsevier 2004. Zbl1076.65093MR2095603
  17. Repin, S., 10.1090/S0025-5718-99-01190-4, Math. Comput. 69(230) (2000), 481-500. Zbl0949.65070MR1681096DOI10.1090/S0025-5718-99-01190-4
  18. Repin, S., A posteriori error estimation for nonlinear variational problems by duality theory., Zapiski Nauchn. Semin. POMI 243 (1997), 201-214. Zbl0904.65064MR1629741
  19. Repin, S., Estimates of deviations from exact solutions of elliptic variational inequalities., Zapiski Nauchn. Semin. POMI 271 (2000), 188-203. Zbl1118.35320MR1810617
  20. Repin, S., A Posteriori Estimates for Partial Differential Equations., Walter de Gruyter, Berlin 2008. Zbl1162.65001MR2458008
  21. Repin, S., Valdman, J., 10.1515/JNUM.2008.003, J. Numer. Math. 16 (2008), 1, 51-81. Zbl1146.65054MR2396672DOI10.1515/JNUM.2008.003
  22. Repin, S., Valdman, J., 10.2478/s11533-009-0035-2, Cent. Eur. J. Math. 7 (2009), 3, 506-519. Zbl1269.74202MR2534470DOI10.2478/s11533-009-0035-2
  23. Ulbrich, M., Semismooth Newton Methods for Variational Inequalities and Constrained Optimization Problems in Function Spaces., SIAM 2011. Zbl1235.49001MR2839219
  24. Valdman, J., Minimization of functional majorant in a posteriori error analysis based on H ( d i v ) multigrid-preconditioned CG method., Advances in Numerical Analysis (2009). Zbl1200.65095MR2739760
  25. Zou, Q., Veeser, A., Kornhuber, R., Gräser, C., 10.1007/s00211-011-0364-5, Numer. Math. 117 (2012), 4, 653-677. Zbl1218.65067MR2776914DOI10.1007/s00211-011-0364-5

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.