Skip to main content
SpringerLink
Log in

Error estimates for the finite element solution of variational inequalities

Part I. Primal theory

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Summary

We analyze the convergence of finite element approximations of some variational inequalities namely the “obstacle problem” and the “unilateral problem”. OptimalO(h) andO(h3/2−∈) error bounds for the obstacle problem (for linear and quadratic elements) and anO(h) error bound for the unilateral problem (with linear elements) are proved.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Baiocchi, C., Pozzi G.A.: An evolution variational inequality related to a diffusion absorption problem, Appl. Math. Optim. (to appear)

  2. Brézis, H.: Problèmes unilateraux. Thése d'etat, Paris, 1971; J. Math. Pures Appl., IX. Sér.72, 1–168 (1971)

  3. Brézis, H.: Nouveaux théorèmes de régularité pour les problèmes unilatéraux. Recontre entre physiciens théoriciens et mathématiciens, Strasbourg 12 (1971)

  4. Brézis, H.: Seuil de régularité pour certains problèmes unilateraux C.R. Acad. Sci. Paris Sér. A273, 35–37 (1971)

    Google Scholar 

  5. Brezzi, F., Sacchi, G.: A finite element approximation of a variational inequality related to hydraulics. To appear

  6. Brézis, H. R., Stampacchia, G.: Sur la régularité de la solution d'inéquations elliptiques. Bull. Soc. Math. France96, 153–180 (1968)

    Google Scholar 

  7. Ciarlet, P.G.: Numerical analysis of the finite element method. Séminaire de Mathématiques Supérieures, Université de Montréal, 16 June–11 July, 1975 (to appear)

  8. Ciarlet, P. G., Raviart, P.A.: General lagrange and hermite interpolation inR″ with applications to finite element methods. Arch. Rational Mech. Anal.46, 177–199 (1972)

    Article  Google Scholar 

  9. Falk, R.: Error estimates for the approximation of a class of variational inequalities. Math. Comput.28, 963–971 (1974)

    Google Scholar 

  10. Falk, R. S., Mercier, B.: Error estimates for elastoplastic problems. R.A.I.R.O. Anal. Num.11, 117–134 (1977)

    Google Scholar 

  11. Frehse, J.: Two dimensional variational problems with thin obstacles. Math. Z.143, 279–288 (1975)

    Google Scholar 

  12. Gagliardo, E.: Proprietà di alcuni classi di funcioni in pui variabili. Ricerche Mat.,7, 102–137 (1958)

    Google Scholar 

  13. Giaquinta, M., Modica, G.: Regolarità lipschitziana per le soluzioni di alcuni problemi di minimo con vincolo. Ann. Mat. Pura Appl., IV. Ser. (to appear)

  14. Glowinski, R.: Introduction to the approximation of elliptic variational inequalities. Report 76006, University of Paris VI, France, 1976

    Google Scholar 

  15. Glowinski, R., Lions, J. L., Trémolieres, R.: Analyse numérique des inéquations variationelles. Paris: Dunod 1976

    Google Scholar 

  16. Hager, W.W.: State constrained convex control problems: Part II. Approximation. Séminaire IRIA, 1975

  17. Hlaváček, I.: Dual finite element analysis for elliptic problems with obstacles on the boundary. I. To appear

  18. Kinderlehrer, D.: How a minimal surface leaves an obstacle. Acta Math.130 221–292 (1973)

    Google Scholar 

  19. Lewy, H., Stampacchia, G.: On the regularity of the solution of a variational inequality. Commun. pure appl. Math.22, 153–188 (1969)

    Google Scholar 

  20. Lions, J. L.: Problèmes aux limites dans les équations aux dérivées partielles. Montreal, Canada: University of Montreal Press 1965

    Google Scholar 

  21. Lions, J.L.: Équations aux dérivées partielles et calcul des variations. Cours de la Faculté des Sciences de Paris, 1967

  22. Lions, J.L., Magenes, E.: Problèmes aux Limites non Homogènes et Applications, tome I. Paris: Dunod 1968

    Google Scholar 

  23. Lions, J.L., Stampacchia, G.: Variational inequalities. Comm. Pure Appl. Math.20, 493–519 (1967)

    Google Scholar 

  24. Mosco, U., Strang, G.: One sided approximation and variational inequalities. Bull. Amer. Math. Soc.80, 308–312 (1974)

    Google Scholar 

  25. Scarpini, F., Vivaldi, M. A.: Error estimates for the approximation of some unilateral problems R.A.I.R.O. Anal. Num.11, 197–208 (1977)

    Google Scholar 

  26. Strang, G.: Approximation in the finite element method. Numer. Math.19, 81–98 (1972)

    Article  Google Scholar 

  27. Strang, G.: The finite element method-linear and nonlinear applications. Proceedings of the International Congress of Mathematicians, Vancouver, Canada, 1974

  28. Strang, G., Berger, A.: The change in solution due to change in domain. Proc. AMS Summer Institute on Partial Differential Equations, Berkeley, 1971

  29. Strang, G., Fix, G.: An analysis of the finite element method. Englewood Cliffs New Jersey: Prentice-Hall 1973

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Politecnico di Torino and Laboratorio di Analisi Numerica del C.N.R., I-27100, Pavia, Italy

    Franco Brezzi

  2. Department of Mathematics, Carnegie-Mellon University, 15213, Pittsburgh, PA, USA

    William W. Hager

  3. Analyse Numérique, Université Pierre et Marie Curie, F-75230, Paris Cedex 05, France

    P. A. Raviart

Authors
  1. Franco Brezzi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. William W. Hager
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. P. A. Raviart
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Supported in part by the Institut de Recherche d'Informatique et d'Automatique and by National Science Foundation grant MCS 75-09457

Rights and permissions

Reprints and permissions

About this article

Cite this article

Brezzi, F., Hager, W.W. & Raviart, P.A. Error estimates for the finite element solution of variational inequalities. Numer. Math. 28, 431–443 (1977). https://doi.org/10.1007/BF01404345

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01404345

Keywords

Search

Navigation