The Obstacle Problem

  • Sören BartelsEmail author
Part of the Springer Series in Computational Mathematics book series (SSCM, volume 47)


The obstacle problem describes constrained deflection of a membrane and serves as a model problem for infinite-dimensional minimization problems with pointwise inequality constraint. Equivalent formulations of the problem including a variational inequality and a characterization with a Lagrange multiplier are derived. The well-posedness and convergence of lowest-order conforming finite element methods, together with a priori and a posteriori error estimates, are established. The application of the semismooth Newton method and a globally convergent primal-dual method together with their implementation conclude the chapter.


  1. 1.
    Brézis, H.R., Stampacchia, G.: Sur la régularité de la solution d’inéquations elliptiques. Bull. Soc. Math. Fr. 96, 153–180 (1968)zbMATHGoogle Scholar
  2. 2.
    Chen, Z., Nochetto, R.H.: Residual type a posteriori error estimates for elliptic obstacle problems. Numer. Math. 84(4), 527–548 (2000).
  3. 3.
    Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. Classics in Applied Mathematics, vol. 40. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2002)CrossRefGoogle Scholar
  4. 4.
    Glowinski, R., Lions, J.L., Trémolières, R.: Numerical Analysis of Variational Inequalities. Studies in Mathematics and Its Applications, vol. 8. North-Holland, Amsterdam (1981)CrossRefzbMATHGoogle Scholar
  5. 5.
    Hintermüller, M., Ito, K., Kunisch, K.: The primal-dual active set strategy as a semismooth Newton method. SIAM J. Optim. 13(3), 865–888 (2003).
  6. 6.
    Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications. Classics in Applied Mathematics, vol. 31. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2000)CrossRefzbMATHGoogle Scholar
  7. 7.
    Rodrigues, J.F.: Obstacle Problems in Mathematical Physics. North-Holland Mathematics Studies, vol. 134. North-Holland, AmsterdamGoogle Scholar
  8. 8.
    Veeser, A.: Efficient and reliable a posteriori error estimators for elliptic obstacle problems. SIAM J. Numer. Anal. 39(1), 146–167 (2001).

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Abteilung für Angewandte MathematikAlbert-Ludwigs-Universität FreiburgFreiburgGermany

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