Foundations of Computational Mathematics

, Volume 12, Issue 3, pp 363–387 | Cite as

Semilinear Mixed Problems on Hilbert Complexes and Their Numerical Approximation

  • Michael Holst
  • Ari SternEmail author


Arnold, Falk, and Winther recently showed (Bull. Am. Math. Soc. 47:281–354, 2010) that linear, mixed variational problems, and their numerical approximation by mixed finite element methods, can be studied using the powerful, abstract language of Hilbert complexes. In another recent article (arXiv:1005.4455), we extended the Arnold–Falk–Winther framework by analyzing variational crimes (à la Strang) on Hilbert complexes. In particular, this gave a treatment of finite element exterior calculus on manifolds, generalizing techniques from surface finite element methods and recovering earlier a priori estimates for the Laplace–Beltrami operator on 2- and 3-surfaces, due to Dziuk (Lecture Notes in Math., vol. 1357:142–155, 1988) and later Demlow (SIAM J. Numer. Anal. 47:805–827, 2009), as special cases. In the present article, we extend the Hilbert complex framework in a second distinct direction: to the study of semilinear mixed problems. We do this, first, by introducing an operator-theoretic reformulation of the linear mixed problem, so that the semilinear problem can be expressed as an abstract Hammerstein equation. This allows us to obtain, for semilinear problems, a priori solution estimates and error estimates that reduce to the Arnold–Falk–Winther results in the linear case. We also consider the impact of variational crimes, extending the results of our previous article to these semilinear problems. As an immediate application, this new framework allows for mixed finite element methods to be applied to semilinear problems on surfaces.


Finite element exterior calculus Mixed finite element methods Semilinear problems Monotone operators 

Mathematics Subject Classification (2010)

65N30 35J91 47H30 


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  1. 1.
    H. Amann, Ein Existenz- und Eindeutigkeitssatz für die Hammersteinsche Gleichung in Banachräumen, Math. Z. 111, 175–190 (1969). MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    H. Amann, Zum Galerkin-Verfahren für die Hammersteinsche Gleichung, Arch. Ration. Mech. Anal. 35, 114–121 (1969). MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    D.N. Arnold, R.S. Falk, R. Winther, Finite element exterior calculus, homological techniques, and applications, Acta Numer. 15, 1–155 (2006). doi: 10.1017/S0962492906210018. MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    D.N. Arnold, R.S. Falk, R. Winther, Finite element exterior calculus: from Hodge theory to numerical stability, Bull., New Ser., Am. Math. Soc. 47(2), 281–354 (2010). doi: 10.1090/S0273-0979-10-01278-4. MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    R.E. Bank, M. Holst, R. Szypowski, Y. Zhu, Finite element error estimates for critical exponent semilinear problems without angle conditions. arXiv:1108.3661 [math.NA] (2011).
  6. 6.
    A. Bossavit, Whitney forms: a class of finite elements for three-dimensional computations in electromagnetism, IEE Proc., A Sci. Meas. Technol. 135(8), 493–500 (1988). Google Scholar
  7. 7.
    F.E. Browder, The solvability of non-linear functional equations, Duke Math. J. 30, 557–566 (1963). MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    F.E. Browder, C.P. Gupta, Monotone operators and nonlinear integral equations of Hammerstein type, Bull. Am. Math. Soc. 75, 1347–1353 (1969). MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    J. Brüning, M. Lesch, Hilbert complexes, J. Funct. Anal. 108(1), 88–132 (1992). doi: 10.1016/0022-1236(92)90147-B. MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    L. Chen, M.J. Holst, J. Xu, The finite element approximation of the nonlinear Poisson–Boltzmann equation, SIAM J. Numer. Anal. 45(6), 2298–2320 (2007). doi: 10.1137/060675514. MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    S.H. Christiansen, Résolution des équations intégrales pour la diffraction d’ondes acoustiques et électromagnétiques: Stabilisation d’algorithmes itératifs et aspects de l’analyse numérique. Ph.D. thesis, École Polytechnique (2002).
  12. 12.
    K. Deckelnick, G. Dziuk, Convergence of a finite element method for non-parametric mean curvature flow, Numer. Math. 72(2), 197–222 (1995). doi: 10.1007/s002110050166. MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    K. Deckelnick, G. Dziuk, Numerical approximation of mean curvature flow of graphs and level sets, in Mathematical Aspects of Evolving Interfaces (Funchal, 2000), Lecture Notes in Math., vol. 1812 (Springer, Berlin, 2003), pp. 53–87. CrossRefGoogle Scholar
  14. 14.
    K. Deckelnick, G. Dziuk, C.M. Elliott, Computation of geometric partial differential equations and mean curvature flow, Acta Numer. 14, 139–232 (2005). doi: 10.1017/S0962492904000224. MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    A. Demlow, Higher-order finite element methods and pointwise error estimates for elliptic problems on surfaces, SIAM J. Numer. Anal. 47(2), 805–827 (2009). doi: 10.1137/070708135. MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    A. Demlow, G. Dziuk, An adaptive finite element method for the Laplace–Beltrami operator on implicitly defined surfaces, SIAM J. Numer. Anal. 45(1), 421–442 (2007) (electronic). doi: 10.1137/050642873. MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    G. Dziuk, Finite elements for the Beltrami operator on arbitrary surfaces, in Partial Differential Equations and Calculus of Variations. Lecture Notes in Math., vol. 1357 (Springer, Berlin, 1988), pp. 142–155. doi: 10.1007/BFb0082865. CrossRefGoogle Scholar
  18. 18.
    G. Dziuk, An algorithm for evolutionary surfaces, Numer. Math. 58(6), 603–611 (1991). doi: 10.1007/BF01385643. MathSciNetzbMATHGoogle Scholar
  19. 19.
    G. Dziuk, C.M. Elliott, Finite elements on evolving surfaces, IMA J. Numer. Anal. 27(2), 262–292 (2007). doi: 10.1093/imanum/drl023. MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    G. Dziuk, J.E. Hutchinson, Finite element approximations to surfaces of prescribed variable mean curvature, Numer. Math. 102(4), 611–648 (2006). doi: 10.1007/s00211-005-0649-7. MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    P.W. Gross, P.R. Kotiuga, Electromagnetic Theory and Computation: A Topological Approach. Mathematical Sciences Research Institute Publications, vol. 48 (Cambridge University Press, Cambridge, 2004). zbMATHCrossRefGoogle Scholar
  22. 22.
    M. Holst, Adaptive numerical treatment of elliptic systems on manifolds, Adv. Comput. Math. 15(1–4), 139–191 (2001). doi: 10.1023/A:1014246117321. MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    M. Holst, G. Nagy, G. Tsogtgerel, Rough solutions of the Einstein constraints on closed manifolds without near-CMC conditions, Commun. Math. Phys. 288, 547–613 (2009). doi: 10.1007/s00220-009-0743-2. MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    M. Holst, A. Stern, Geometric variational crimes: Hilbert complexes, finite element exterior calculus, and problems on hypersurfaces. arXiv:1005.4455 [math.NA] (2010).
  25. 25.
    G.J. Minty, Monotone (nonlinear) operators in Hilbert space, Duke Math. J. 29, 341–346 (1962). MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    J.C. Nédélec, Curved finite element methods for the solution of singular integral equations on surfaces in ℝ3, Comput. Methods Appl. Mech. Eng. 8(1), 61–80 (1976). zbMATHCrossRefGoogle Scholar
  27. 27.
    J.C. Nédélec, Mixed finite elements in ℝ3, Numer. Math. 35(3), 315–341 (1980). doi: 10.1007/BF01396415. MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    J.C. Nédélec, A new family of mixed finite elements in ℝ3, Numer. Math. 50(1), 57–81 (1986). doi: 10.1007/BF01389668. MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    I. Stakgold, M. Holst, Green’s Functions and Boundary Value Problems, Pure and Applied Mathematics (Hoboken), 3rd edn. (Wiley, Hoboken, 2011). zbMATHCrossRefGoogle Scholar
  30. 30.
    E. Zeidler, Nonlinear Functional Analysis and Its Applications, Part II/B: Nonlinear Monotone Operators (Springer, New York, 1990). Translated from the German by the author and Leo F. Boron. CrossRefGoogle Scholar

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© SFoCM 2011

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California, San DiegoLa JollaUSA

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