Kybernetika 49 no. 5, 738-754, 2013

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

Petr Harasim and Jan Valdman

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.

Keywords:

finite element method, obstacle problem, a posteriori error estimate, functional majorant, variational inequalities, Uzawa algorithm

Classification:

34B15, 65K15, 65L60, 74K05, 74M15, 74S05

paper.pdf

References:

  1. M. Ainsworth and J. T. Oden: A Posteriori Error Estimation in Finite Element Analysis. Wiley and Sons, New York 2000.   CrossRef
  2. I. Babu{š}ka and T. Strouboulis: The Finite Element Method and its Reliability. Oxford University Press, New York 2001.   CrossRef
  3. W. Bangerth and R. Rannacher: Adaptive Finite Element Methods for Differential Equations. Birkhäuser, Berlin 2003.   CrossRef
  4. D. Braess, R. H. W. Hoppe and J. Schöberl: A posteriori estimators for obstacle problems by the hypercircle method. Comp. Visual. Sci. 11 (2008), 351-362.   CrossRef
  5. F. Brezi, W. W. Hager and P. A. Raviart: Error estimates for the finite element solution of variational inequalities I. Numer. Math. 28 (1977), 431-443.   CrossRef
  6. H. Buss and S. Repin: 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.   CrossRef
  7. C. Carstensen and C. Merdon: A posteriori error estimator completition for conforming obstacle problems. Numer. Methods Partial Differential Equations 29 (2013), 667-�692.   CrossRef
  8. Z. Dostál: Optimal Quadratic Programming Algorithms. Springer 2009.   CrossRef
  9. R. S. Falk: Error estimates for the approximation of a class of variational inequalities. Math. Comput. 28 (1974), 963-971.   CrossRef
  10. M. Fuchs and S. Repin: A Posteriori Error Estimates for the Approximations of the Stresses in the Hencky Plasticity Problem. Numer. Funct. Anal. Optim. 32 (2011), 610-640.   CrossRef
  11. R. Glowinski, J. L. Lions and R. Trémolieres: Numerical Analysis of Variational Inequalities. North-Holland 1981.   CrossRef
  12. I. Hlaváček, J. Haslinger, J. Nečas and J. Lovíšek: Solution of Variational Inequalities in Mechanics. Applied Mathematical Sciences 66, Springer-Verlag, New York 1988.   CrossRef
  13. N. Kikuchi and J. T. Oden: Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods. SIAM 1995.   CrossRef
  14. J. Kraus and S. Tomar: Algebraic multilevel iteration method for lowest-order Raviart-Thomas space and applications. Internat. J. Numer. Meth. Engrg. 86 (2011), 1175-1196.   CrossRef
  15. J. L. Lions and G. Stampacchia: Variational inequalities. Comm. Pure Appl. Math. XX(3) (1967), 493-519.   CrossRef
  16. P. Neittaanmäki and S. Repin: Reliable Methods for Computer Simulation (Error Control and a Posteriori Estimates). Elsevier 2004.   CrossRef
  17. S. Repin: A posteriori error estimation for variational problems with uniformly convex functionals. Math. Comput. 69(230) (2000), 481-500.   CrossRef
  18. S. Repin: A posteriori error estimation for nonlinear variational problems by duality theory. Zapiski Nauchn. Semin. POMI 243 (1997), 201-214.   CrossRef
  19. S. Repin: Estimates of deviations from exact solutions of elliptic variational inequalities. Zapiski Nauchn. Semin. POMI 271 (2000), 188-203.   CrossRef
  20. S. Repin: A Posteriori Estimates for Partial Differential Equations. Walter de Gruyter, Berlin 2008.   CrossRef
  21. S. Repin and J. Valdman: Functional a posteriori error estimates for problems with nonlinear boundary conditions. J. Numer. Math. 16 (2008), 1, 51-81.   CrossRef
  22. S. Repin and J. Valdman: Functional a posteriori error estimates for incremental models in elasto-plasticity. Cent. Eur. J. Math. 7 (2009), 3, 506-519.   CrossRef
  23. M. Ulbrich: Semismooth Newton Methods for Variational Inequalities and Constrained Optimization Problems in Function Spaces. SIAM 2011.   CrossRef
  24. J. Valdman: Minimization of functional majorant in a posteriori error analysis based on $H(div)$ multigrid-preconditioned CG method. Advances in Numerical Analysis (2009).   CrossRef
  25. Q. Zou, A. Veeser, R. Kornhuber and C. Gräser: Hierarchical error estimates for the energy functional in obstacle problems. Numer. Math. 117 (2012), 4, 653-677.   CrossRef