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On the Use of Elliptic Regularity Theory for the Numerical Solution of Variational Problems

  • Axel Dreves
  • Joachim Gwinner
  • Nina OvcharovaEmail author
Chapter
Part of the Springer Optimization and Its Applications book series (SOIA, volume 113)

Abstract

In this article we show the crucial role of elliptic regularity theory for the development of efficient numerical methods for the solution of some variational problems. Here we focus on a class of elliptic multiobjective optimal control problems that can be formulated as jointly convex generalized Nash equilibrium problems (GNEPs) and on nonsmooth boundary value problems that stem from contact mechanics leading to elliptic variational inequalities (VIs).

Keywords

Complementarity problem Dual mixed formulation Elliptic boundary value problem Jointly convex generalized Nash equilibrium problem Lagrange multiplier Multiobjective optimal control Normalized Nash equilibrium Obstacle problem Saddle point formulation Signorini problem Smooth domain Unilateral contact Variational inequality 

AMS

90C29 90C33 49J21 49N60 

References

  1. 1.
    R.A. Adams, J.J.F. Fournier, Sobolev Spaces (Elsevier, Amsterdam, 2003)zbMATHGoogle Scholar
  2. 2.
    K.J. Arrow, G. Debreu, Existence of an equilibrium for a competitive economy. Econometrica 22, 265–290 (1954)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    J.-P. Aubin, Approximation of Elliptic Boundary-Value Problems. Pure and Applied Mathematics, vol. XXVI (Wiley-Interscience, New York, 1972)Google Scholar
  4. 4.
    I. Babuška, G.N. Gatica, On the mixed finite element method with Lagrange multipliers. Numer. Methods Partial Differ. Equ. 19 (2), 192–210 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    C. Bacuta, J.H. Bramble, Regularity estimates for solutions of the equations of linear elasticity in convex plane polygonal domains. Z. Angew. Math. Phys. 54 (5), 874–878 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    L. Banz, A. Schröder, Biorthogonal basis functions in hp-adaptive FEM for elliptic obstacle problems. Comput. Math. Appl. 70 (8), 1721–1742 (2015)MathSciNetCrossRefGoogle Scholar
  7. 7.
    M. Barboteu, K. Bartosz, P. Kalita, An analytical and numerical approach to a bilateral contact problem with nonmonotone friction. Int. J. Appl. Math. Comput. Sci. 23 (2), 263–276 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    J.M. Borwein, A.S. Lewis, Partially finite convex programming, Part I: quasi relative interiors and duality theory. Math. Programm. 57 (1–3), 15–48 (1992)zbMATHGoogle Scholar
  9. 9.
    R.I. Boţ, E.R. Csetnek, A. Moldovan, Revisiting some duality theorems via the quasirelative interior in convex optimization. J. Optim. Theory Appl. 139 (1), 67–84 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    D. Braess, Finite Elements (Cambridge University Press, Cambridge, 2007)CrossRefzbMATHGoogle Scholar
  11. 11.
    F. Brezzi, M. Fortin, Mixed and Hybrid Finite Element Methods (Springer, New York, 1991)CrossRefzbMATHGoogle Scholar
  12. 12.
    J. Czepiel, P. Kalita, Numerical solution of a variational-hemivariational inequality modelling simplified adhesion of an elastic body. IMA J. Numer. Anal. 35 (1), 372–393 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    P. Daniele, S. Giuffrè, General infinite dimensional duality and applications to evolutionary network equilibrium problems. Optim. Lett. 1 (3), 227–243 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    P. Daniele, S. Giuffrè, G. Idone, A. Maugeri, Infinite dimensional duality and applications. Math. Ann. 339 (1), 221–239 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    P. Daniele, S. Giuffrè, A. Maugeri, F. Raciti, Duality theory and applications to unilateral problems. J. Optim. Theory Appl. 162 (3), 718–734 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Z. Ding, A proof of the trace theorem of Sobolev spaces on Lipschitz domains. Proc. Am. Math. Soc. 124 (2), 591–600 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    A. Dreves, J. Gwinner, Jointly convex generalized Nash equilibria and elliptic multiobjective optimal control. J. Optim. Theory Appl. 168 (3), 1065–1086 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    R.S. Falk, Error estimates for the approximation of a class of variational inequalities. Math. Comput. 28, 963–971 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    G.N. Gatica, N. Heuer, S. Meddahi, On the numerical analysis of nonlinear twofold saddle point problems. IMA J. Numer. Anal. 23 (2), 301–330 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order. Classics in Mathematics (Springer, Berlin, 2001)Google Scholar
  21. 21.
    R. Glowinski, Numerical Methods for Nonlinear Variational Problems (Springer, Berlin, 2008)zbMATHGoogle Scholar
  22. 22.
    P. Grisvard, Elliptic Problems in Nonsmooth Domains. Classics in Applied Mathematics, vol. 69 (SIAM, Philadelphia, PA, 2011)Google Scholar
  23. 23.
    J. Gwinner, An extension lemma and homogeneous programming. J. Optim. Theory Appl. 47 (3), 321–336 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    J. Gwinner, On the p-version approximation in the boundary element method for a variational inequality of the second kind modelling unilateral contact and given friction. Appl. Numer. Math. 59 (11), 2774–2784 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    J. Gwinner, hp-FEM convergence for unilateral contact problems with Tresca friction in plane linear elastostatics. J. Comput. Appl. Math. 254, 175–184 (2013)Google Scholar
  26. 26.
    J. Gwinner, Three-field modelling of nonlinear nonsmooth boundary value problems and stability of differential mixed variational inequalities. Abstr. Appl. Anal. (2013). doi:http://dx.doi.org/10.1155/2013/108043. ID 108043
  27. 27.
    J. Gwinner, Multi-field modeling of nonsmooth problems of continuum mechanics,differential mixed variational inequalities and their stability, in Applied Mathematics in Tunisia. Springer Proceedings of Mathematical Statistics, vol. 131 (Springer, Cham, 2015), pp. 119–139Google Scholar
  28. 28.
    J. Gwinner, N. Ovcharova, From solvability and approximation of variational inequalities to solution of nondifferentiable optimization problems in contact mechanics. Optimization 64 (8), 1683–1702 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    M. Hintermüller, V.A. Kovtunenko, K. Kunisch, Obstacle problems with cohesion: a hemivariational inequality approach and its efficient numerical solution. SIAM J. Optim. 21 (2), 491–516 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    M. Hintermüller, T. Surowiec, A PDE-constrained generalized Nash equilibrium problem with pointwise control and state constraints. Pac. J. Optim. 9 (2), 251–273 (2013)MathSciNetzbMATHGoogle Scholar
  31. 31.
    M. Hintermüller, T. Surowiec, A. Kämmler, Generalized Nash equilibrium problems in banach spaces: theory, Nikaido–Isoda-based path-following methods, and applications. SIAM J. Optim. 25 (3), 1826–1856 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    J. Jahn, Introduction to the Theory of Nonlinear Optimization (Springer, Berlin, 1996)CrossRefzbMATHGoogle Scholar
  33. 33.
    R. Jensen, Boundary regularity for variational inequalities. Indiana Univ. Math. J. 29 (4), 495–504 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    V. Jeyakumar, H. Wolkowicz, Generalizations of Slater’s constraint qualification for infinite convex programs. Math. Programm. Ser. B 57 (1), 85–101 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    J. Kadlec, The regularity of the solution of the Poisson problem in a domain whose boundary is similar to that of a convex domain. Czechoslovak Math. J. 14 (89), 386–393 (1964)MathSciNetzbMATHGoogle Scholar
  36. 36.
    N. Kikuchi, J.T. Oden, Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods (SIAM, Philadelphia, PA, 1988)CrossRefzbMATHGoogle Scholar
  37. 37.
    D. Kinderlehrer, Remarks about Signorini’s problem in linear elasticity. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 8 (4), 605–645 (1981)MathSciNetzbMATHGoogle Scholar
  38. 38.
    D. Kinderlehrer, G. Stampacchia, An Introduction to Variational Inequalities and Their Applications (SIAM, Philadelphia, PA, 2000)CrossRefzbMATHGoogle Scholar
  39. 39.
    D. Knees, A. Schröder, Global spatial regularity for elasticity models with cracks, contact and other nonsmooth constraints. Math. Methods Appl. Sci. 35 (15), 1859–1884 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    A. Kufner, A.-M. Sändig, Some Applications of Weighted Sobolev Spaces (Teubner, Leipzig, 1987)zbMATHGoogle Scholar
  41. 41.
    B.P. Lamichhane, B.I. Wohlmuth, Biorthogonal bases with local support and approximation properties. Math. Comput. 76 (257), 233–249 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    D. Mitrea, M. Mitrea, L. Yan, Boundary value problems for the Laplacian in convex and semiconvex domains. J. Funct. Anal. 258 (8), 2507–2585 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    H. Nikaidô, K. Isoda, Note on non-cooperative convex games. Pac. J. Math. 5, 807–815 (1955)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    M.A. Noor, K.I. Noor, T.M. Rassias, Some aspects of variational inequalities. J. Comput. Appl. Math. 47 (3), 285–312 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    N. Ovcharova, J. Gwinner, A study of regularization techniques of nondifferentiable optimization in view of application to hemivariational inequalities. J. Optim. Theory Appl. 162 (3), 754–778 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    R.T. Rockafellar, Conjugate Duality and Optimization (SIAM, Philadelphia, PA, 1974)CrossRefzbMATHGoogle Scholar
  47. 47.
    R. Schumann, Regularity for Signorini’s problem in linear elasticity. Manuscripta Math. 63 (3), 255–291 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    R. Schumann, Regularity for variational inequalities—a survey of results, in From Convexity to Nonconvexity. Nonconvex Optimization and its Applications, vol. 55 (Kluwer Academic Publishers, Dordrecht, 2001), pp. 269–282Google Scholar
  49. 49.
    G.E. Stavroulakis, E.S. Mistakidis, Numerical treatment of hemivariational inequalities in mechanics: two methods based on the solution of convex subproblems. Comput. Mech. 16 (6), 406–416 (1995)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of Aerospace EngineeringUniversität der Bundeswehr MünchenMünchenGermany

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