Sharp upper global a posteriori error estimates for nonlinear elliptic variational problems

János Karátson; Sergey Korotov

Applications of Mathematics (2009)

  • Volume: 54, Issue: 4, page 297-336
  • ISSN: 0862-7940

Abstract

top
The paper is devoted to the problem of verification of accuracy of approximate solutions obtained in computer simulations. This problem is strongly related to a posteriori error estimates, giving computable bounds for computational errors and detecting zones in the solution domain where such errors are too large and certain mesh refinements should be performed. A mathematical model embracing nonlinear elliptic variational problems is considered in this work. Based on functional type estimates developed on an abstract level, we present a general technology for constructing computable sharp upper bounds for the global error for various particular classes of elliptic problems. Here the global error is understood as a suitable energy type difference between the true and computed solutions. The estimates obtained are completely independent of the numerical technique used to obtain approximate solutions, and are sharp in the sense that they can be, in principle, made as close to the true error as resources of the used computer allow. The latter can be achieved by suitably tuning the auxiliary parameter functions, involved in the proposed upper error bounds, in the course of the calculations.

How to cite

top

Karátson, János, and Korotov, Sergey. "Sharp upper global a posteriori error estimates for nonlinear elliptic variational problems." Applications of Mathematics 54.4 (2009): 297-336. <http://eudml.org/doc/37823>.

@article{Karátson2009,
abstract = {The paper is devoted to the problem of verification of accuracy of approximate solutions obtained in computer simulations. This problem is strongly related to a posteriori error estimates, giving computable bounds for computational errors and detecting zones in the solution domain where such errors are too large and certain mesh refinements should be performed. A mathematical model embracing nonlinear elliptic variational problems is considered in this work. Based on functional type estimates developed on an abstract level, we present a general technology for constructing computable sharp upper bounds for the global error for various particular classes of elliptic problems. Here the global error is understood as a suitable energy type difference between the true and computed solutions. The estimates obtained are completely independent of the numerical technique used to obtain approximate solutions, and are sharp in the sense that they can be, in principle, made as close to the true error as resources of the used computer allow. The latter can be achieved by suitably tuning the auxiliary parameter functions, involved in the proposed upper error bounds, in the course of the calculations.},
author = {Karátson, János, Korotov, Sergey},
journal = {Applications of Mathematics},
keywords = {a posteriori error estimation; error control in energy norm; error estimates of functional type; elliptic equation of second order; elliptic equation of fourth order; second order elasticity system; mixed boundary conditions; gradient averaging; error control in energy norm; error estimates of functional type; elliptic equation of second order; elliptic equation of fourth order; nonlinear elliptic variational problems},
language = {eng},
number = {4},
pages = {297-336},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Sharp upper global a posteriori error estimates for nonlinear elliptic variational problems},
url = {http://eudml.org/doc/37823},
volume = {54},
year = {2009},
}

TY - JOUR
AU - Karátson, János
AU - Korotov, Sergey
TI - Sharp upper global a posteriori error estimates for nonlinear elliptic variational problems
JO - Applications of Mathematics
PY - 2009
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 54
IS - 4
SP - 297
EP - 336
AB - The paper is devoted to the problem of verification of accuracy of approximate solutions obtained in computer simulations. This problem is strongly related to a posteriori error estimates, giving computable bounds for computational errors and detecting zones in the solution domain where such errors are too large and certain mesh refinements should be performed. A mathematical model embracing nonlinear elliptic variational problems is considered in this work. Based on functional type estimates developed on an abstract level, we present a general technology for constructing computable sharp upper bounds for the global error for various particular classes of elliptic problems. Here the global error is understood as a suitable energy type difference between the true and computed solutions. The estimates obtained are completely independent of the numerical technique used to obtain approximate solutions, and are sharp in the sense that they can be, in principle, made as close to the true error as resources of the used computer allow. The latter can be achieved by suitably tuning the auxiliary parameter functions, involved in the proposed upper error bounds, in the course of the calculations.
LA - eng
KW - a posteriori error estimation; error control in energy norm; error estimates of functional type; elliptic equation of second order; elliptic equation of fourth order; second order elasticity system; mixed boundary conditions; gradient averaging; error control in energy norm; error estimates of functional type; elliptic equation of second order; elliptic equation of fourth order; nonlinear elliptic variational problems
UR - http://eudml.org/doc/37823
ER -

References

top
  1. Ainsworth, M., Oden, J. T., A Posteriori Error Estimation in Finite Element Analysis, John Wiley & Sons (2000). (2000) Zbl1008.65076MR1885308
  2. Axelsson, O., Padiy, A., 10.1002/1097-0207(20001230)49:12<1479::AID-NME4>3.0.CO;2-4, Int. J. Numer. Methods Eng. 49 (2000), 1479-1493. (2000) Zbl0994.74066MR1797720DOI10.1002/1097-0207(20001230)49:12<1479::AID-NME4>3.0.CO;2-4
  3. Axelsson, O., Maubach, J., 10.1016/0045-7825(88)90095-3, Comput. Methods Appl. Mech. Eng. 71 (1988), 41-67. (1988) MR0967153DOI10.1016/0045-7825(88)90095-3
  4. Babuška, I., Strouboulis, T., The Finite Element Method and Its Reliability, Oxford University Press New York (2001). (2001) MR1857191
  5. Bangerth, W., Rannacher, R., Adaptive Finite Element Methods for Differential Equations. Lectures in Mathematics ETH Zürich, Birkhäuser Basel (2003). (2003) MR1960405
  6. Becker, R., Rannacher, R., A feed-back approach to error control in finite element methods: Basic approach and examples, East-West J. Numer. Math. 4 (1996), 237-264. (1996) MR1430239
  7. Blaheta, R., Multilevel Newton methods for nonlinear problems with applications to elasticity, Copernicus 940820. Technical report. 
  8. Brandts, J., Křížek, M., 10.1093/imanum/23.3.489, IMA J. Numer. Anal. 23 (2003), 489-505. (2003) Zbl1042.65081MR1987941DOI10.1093/imanum/23.3.489
  9. Carstensen, C., Funken, S. A., Constants in Clément-interpolation error and residual based a posteriori error estimates in finite element methods, East-West J. Numer. Math. 8 (2000), 153-175. (2000) Zbl0973.65091MR1807259
  10. Ciarlet, Ph. G., The Finite Element Method for Elliptic Problems. Studies in Mathematics and Its Applications Vol. 4, North-Holland Publishing Amsterdam-New York-Oxford (1978). (1978) MR0520174
  11. Concus, P., 10.1016/0021-9991(67)90043-5, J. Comput. Phys. 1 (1967), 330-342. (1967) Zbl0154.41103DOI10.1016/0021-9991(67)90043-5
  12. Eriksson, K., Estep, D., Hansbo, P., Johnson, C., 10.1017/S0962492900002531, Acta Numerica 1995 Cambridge University Press Cambridge (1995), 105-158. (1995) Zbl0829.65122MR1352472DOI10.1017/S0962492900002531
  13. Faragó, I., Karátson, J., Numerical Solution of Nonlinear Elliptic Problems via Preconditioning Operators. Theory and Applications. Advances in Computation, Vol. 11, NOVA Science Publishers New York (2002). (2002) MR2106499
  14. Frolov, M. E., 10.1515/1569398042395970, Russ. J. Numer. Anal. Math. Model. 19 (2004), 407-418. (2004) Zbl1064.65129MR2107584DOI10.1515/1569398042395970
  15. Glowinski, R., Marrocco, A., 10.1016/0045-7825(74)90042-5, Computer Methods Appl. Mech. Engin. 3 (1974), 55-85. (1974) Zbl0288.65068MR0413547DOI10.1016/0045-7825(74)90042-5
  16. Grisvard, P., Elliptic Problems in Nonsmooth Domains, Pitman Boston-London-Melbourne (1985). (1985) Zbl0695.35060MR0775683
  17. Han, W., A Posteriori Error Analysis via Duality Theory. With Applications in Modeling and Numerical Approximations. Advances in Mechanics and Mathematics Vol. 8, Springer New York (2005). (2005) MR2101057
  18. Hannukainen, A., Finite Element Methods for Maxwell's Equations. Master Thesis, Institute of Mathematics, Helsinki University of Technology Helsinki (2007). (2007) 
  19. Hannukainen, A., Korotov, S., Techniques for a posteriori error estimation in terms of linear functionals for elliptic type boundary value problems, Far East J. Appl. Math. 21 (2005), 289-304. (2005) Zbl1092.65097MR2216003
  20. Hlaváček, I., Chleboun, J., Babuška, I., Uncertain Input Data Problems and the Worst Scenario Method, Elsevier (2004). (2004) Zbl1116.74003MR2285091
  21. Hlaváček, I., Křížek, M., On a superconvergent finite element scheme for elliptic systems. I. Dirichlet boundary condition; II. Boundary conditions of Newton's or Neumann's type; III. Optimal interior estimates, Apl. Mat. 32 (1987), 131-154; 200-213; 276-289. (1987) MR0885758
  22. Horgan, C. O., 10.1137/1037123, SIAM Rev. 37 (1995), 491-511. (1995) Zbl0840.73010MR1368384DOI10.1137/1037123
  23. Kachanov, L. M., Foundations of the Theory of Plasticity, North-Holland Amstedram (1971). (1971) Zbl0231.73015MR0483881
  24. Karátson, J., 10.1016/j.jmaa.2008.05.053, J. Math. Anal. Appl. 346 (2008), 170-176. (2008) Zbl1152.47047MR2428281DOI10.1016/j.jmaa.2008.05.053
  25. Karátson, J., Korotov, S., Discrete maximum principles for FEM solutions of some nonlinear elliptic interface problems, Int. J. Numer. Anal. Model. 6 (2009), 1-16. (2009) Zbl1163.65076MR2574894
  26. Karátson, J., Korotov, S., Sharp upper global a posteriori error estimates for nonlinear elliptic variational problems, Helsinki University of Technology, Institute of Mathematics, Research Report A527; July 2007. 
  27. Korotov, S., 10.1016/j.cam.2005.06.038, J. Comput. Appl. Math. 191 (2006), 216-227. (2006) Zbl1089.65120MR2219926DOI10.1016/j.cam.2005.06.038
  28. Korotov, S., 10.1007/s10492-007-0012-7, Appl. Math. 52 (2007), 235-249. (2007) Zbl1164.65485MR2316154DOI10.1007/s10492-007-0012-7
  29. Korotov, S., 10.1016/j.apm.2007.04.013, Appl. Math. Modelling 32 (2008), 1579-1586. (2008) Zbl1176.65126MR2412433DOI10.1016/j.apm.2007.04.013
  30. Křížek, M., Neittaanmäki, P., Mathematical and Numerical Modelling in Electrical Engineering: Theory and Applications, Kluwer Academic Publishers Dordrecht (1996). (1996) MR1431889
  31. Kuzmin, D., Hannukainen, A., Korotov, S., 10.1016/j.cam.2007.04.033, J. Comput. Appl. Math. 218 (2008), 70-78. (2008) MR2431599DOI10.1016/j.cam.2007.04.033
  32. Maz'ja, V. G., Sobolev Spaces, Springer Berlin (1985). (1985) Zbl0692.46023
  33. Miersemann, E., 10.4171/ZAA/29, Z. Anal. Anw. 1 (1982), 59-71. (1982) Zbl0518.35011MR0719164DOI10.4171/ZAA/29
  34. Mikhlin, S. G., The Numerical Performance of Variational Methods. Walters Noordhoff Series of Monographs and Textbooks on Pure and Applied Mathematics, Walters Noordhoff Publishing Groningen (1971). (1971) MR0278506
  35. Mikhlin, S. G., Constants in Some Inequalities of Analysis. A Wiley-Interscience Publication, John Wiley & Sons Chichester (1986). (1986) MR0853915
  36. Muzalevsky, A. V., Repin, S. I., 10.1515/156939803322008209, Russ. J. Numer. Anal. Math. Model. 18 (2003), 65-85. (2003) Zbl1027.74070MR1961986DOI10.1515/156939803322008209
  37. Nečas, J., Les méthodes directes en théorie des équations elliptiques, Academia Prague (1967). (1967) MR0227584
  38. Nečas, J., Hlaváček, I., Mathematical Theory of Elastic and Elasto-Plastic Bodies: An Introduction. Studies in Applied Mechanics 3, Elsevier Scientific Publishing Amsterdam-New York (1980). (1980) MR0600655
  39. Neittaanmäki, P., Repin, S., A posteriori error estimates for boundary-value problems related to the biharmonic operator, East-West J. Numer. Math. 9 (2001), 157-178. (2001) Zbl0986.65101MR1836871
  40. Neittaanmäki, P., Repin, S., Reliable Methods for Computer Simulation. Error Control and A Posteriori Estimates. Studies in Mathematics and Its Applications 33, Elsevier Science B.V. Amsterdam (2004). (2004) MR2095603
  41. Neumaier, A., 10.1016/j.cam.2007.04.037, J. Comput. Appl. Math. 218 (2008), 125-136. (2008) Zbl1145.65088MR2431605DOI10.1016/j.cam.2007.04.037
  42. Oden, J. T., Prudhomme, S., 10.1016/S0898-1221(00)00317-5, Comput. Math. Appl. 41 (2001), 735-756. (2001) Zbl0987.65110MR1822600DOI10.1016/S0898-1221(00)00317-5
  43. Repin, S., A posteriori error estimation for nonlinear variational problems by duality theory, Zap. Nauchn. Semin. St. Petersburg, Otdel. Mat. Inst. Steklov (POMI) 243 (1997), 201-214. (1997) Zbl0904.65064MR1629741
  44. Repin, S., Sauter, S., Smolianski, A., 10.1007/s00607-003-0013-7, Computing 70 (2003), 205-233. (2003) Zbl1128.35319MR2011610DOI10.1007/s00607-003-0013-7
  45. Repin, S., Sauter, S., Smolianski, A., 10.1016/S0377-0427(03)00491-6, J. Comput. Appl. Math. 164/165 (2004), 601-612. (2004) Zbl1038.65114MR2056902DOI10.1016/S0377-0427(03)00491-6
  46. Repin, S., Xanthis, L. S., 10.1016/S0045-7825(96)01136-X, Comput. Methods Appl. Mech. Eng. 138 (1996), 317-339. (1996) MR1422336DOI10.1016/S0045-7825(96)01136-X
  47. Rüter, M., Korotov, S., Steenbock, C., 10.1007/s00466-006-0069-2, Comput. Mech. 39 (2007), 787-797. (2007) Zbl1178.74172MR2298591DOI10.1007/s00466-006-0069-2
  48. Vejchodský, T., 10.1093/imanum/dri043, IMA J. Numer. Anal. 26 (2006), 525-540. (2006) Zbl1096.65112MR2241313DOI10.1093/imanum/dri043
  49. Verfürth, R., A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques, John Wiley & Sons-Teubner Chichester-Stuttgart (1996). (1996) 
  50. Vohralík, M., 10.1016/j.crma.2008.03.006, C. R. Math., Acad. Sci. Paris 346 (2008), 687-690. (2008) Zbl1142.65086MR2423279DOI10.1016/j.crma.2008.03.006
  51. Waterhouse, W. C., The absolute-value estimate for symmetric multilinear forms, Linear Algebra Appl. 128 (1990), 97-105. (1990) Zbl0699.15012MR1049074
  52. Zeidler, E., Nonlinear Functional Analysis and Its Applications, Springer New York (1986). (1986) Zbl0583.47050MR0816732
  53. Zienkiewicz, O. C., Zhu, J. Z., 10.1002/nme.1620240206, Int. J. Numer. Methods Eng. 24 (1987), 337-357. (1987) Zbl0602.73063MR0875306DOI10.1002/nme.1620240206

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.