Bubbles Enriched Quadratic Finite Element Method for the 3D-Elliptic Obstacle Problem

Sharat Gaddam 1  and Thirupathi Gudi 2
  • 1 Department of Mathematics, Indian Institute of Science, 560012, Bangalore, India
  • 2 Department of Mathematics, Indian Institute of Science, 560012, Bangalore, India
Sharat Gaddam and Thirupathi Gudi

Abstract

An optimally convergent (with respect to the regularity) quadratic finite element method for the two-dimensional obstacle problem on simplicial meshes is studied in []. There was no analogue of a quadratic finite element method on tetrahedron meshes for the three-dimensional obstacle problem. In this article, a quadratic finite element enriched with element-wise bubble functions is proposed for the three-dimensional elliptic obstacle problem. A priori error estimates are derived to show the optimal convergence of the method with respect to the regularity. Further, a posteriori error estimates are derived to design an adaptive mesh refinement algorithm. A numerical experiment illustrating the theoretical result on a priori error estimates is presented.

  • [1]

    M. Ainsworth and J. T. Oden, A Posteriori Error Estimation in Finite Element Analysis, Pure Appl. Math. (New York), Wiley-Interscience, New York, 2000.

  • [2]

    M. Ainsworth, J. T. Oden and C.-Y. Lee, Local a posteriori error estimators for variational inequalities, Numer. Methods Partial Differential Equations 9 (1993), no. 1, 23–33.

    • Crossref
    • Export Citation
  • [3]

    K. Atkinson and W. Han, Theoretical Numerical Analysis, 3rd ed., Texts Appl. Math. 39, Springer, New York, 2009.

  • [4]

    L. Banz and E. P. Stephan, A posteriori error estimates of hp-adaptive IPDG-FEM for elliptic obstacle problems, Appl. Numer. Math. 76 (2014), 76–92.

    • Crossref
    • Export Citation
  • [5]

    S. Bartels and C. Carstensen, Averaging techniques yield reliable a posteriori finite element error control for obstacle problems, Numer. Math. 99 (2004), no. 2, 225–249.

    • Crossref
    • Export Citation
  • [6]

    F. Ben Belgacem, Numerical simulation of some variational inequalities arisen from unilateral contact problems by the finite element methods, SIAM J. Numer. Anal. 37 (2000), no. 4, 1198–1216.

    • Crossref
    • Export Citation
  • [7]

    H. Blum and F.-T. Suttmeier, An adaptive finite element discretisation for a simplified Signorini problem, Calcolo 37 (2000), no. 2, 65–77.

    • Crossref
    • Export Citation
  • [8]

    D. Braess, A posteriori error estimators for obstacle problems—Another look, Numer. Math. 101 (2005), no. 3, 415–421.

    • Crossref
    • Export Citation
  • [9]

    D. Braess, C. Carstensen and R. H. W. Hoppe, Convergence analysis of a conforming adaptive finite element method for an obstacle problem, Numer. Math. 107 (2007), no. 3, 455–471.

    • Crossref
    • Export Citation
  • [10]

    D. Braess, C. Carstensen and R. H. W. Hoppe, Error reduction in adaptive finite element approximations of elliptic obstacle problems, J. Comput. Math. 27 (2009), no. 2–3, 148–169.

  • [11]

    S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, 3rd ed., Texts Appl. Math. 15, Springer, New York, 2008.

  • [12]

    S. C. Brenner, L. Sung and Y. Zhang, Finite element methods for the displacement obstacle problem of clamped plates, Math. Comp. 81 (2012), no. 279, 1247–1262.

    • Crossref
    • Export Citation
  • [13]

    S. C. Brenner, L.-Y. Sung, H. Zhang and Y. Zhang, A quadratic C 0 C^{0} interior penalty method for the displacement obstacle problem of clamped Kirchhoff plates, SIAM J. Numer. Anal. 50 (2012), no. 6, 3329–3350.

    • Crossref
    • Export Citation
  • [14]

    F. Brezzi, W. W. Hager and P.-A. Raviart, Error estimates for the finite element solution of variational inequalities, Numer. Math. 28 (1977), no. 4, 431–443.

    • Crossref
    • Export Citation
  • [15]

    Z. Chen and R. H. Nochetto, Residual type a posteriori error estimates for elliptic obstacle problems, Numer. Math. 84 (2000), no. 4, 527–548.

    • Crossref
    • Export Citation
  • [16]

    P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland Publishing, Amsterdam, 1978.

  • [17]

    G. Drouet and P. Hild, Optimal convergence for discrete variational inequalities modelling Signorini contact in 2D and 3D without additional assumptions on the unknown contact set, SIAM J. Numer. Anal. 53 (2015), no. 3, 1488–1507.

    • Crossref
    • Export Citation
  • [18]

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

    • Crossref
    • Export Citation
  • [19]

    M. Feischl, M. Page and D. Praetorius, Convergence of adaptive FEM for some elliptic obstacle problem with inhomogeneous Dirichlet data, Int. J. Numer. Anal. Model. 11 (2014), no. 1, 229–253.

  • [20]

    R. Glowinski, Numerical Methods for Nonlinear Variational Problems. Reprint of the 1984 original, Sci. Comput., Springer, Berlin, 2008.

  • [21]

    T. Gudi and K. Porwal, A posteriori error control of discontinuous Galerkin methods for elliptic obstacle problems, Math. Comp. 83 (2014), no. 286, 579–602.

  • [22]

    T. Gudi and K. Porwal, A remark on the a posteriori error analysis of discontinuous Galerkin methods for the obstacle problem, Comput. Methods Appl. Math. 14 (2014), no. 1, 71–87.

  • [23]

    T. Gudi and K. Porwal, A reliable residual based a posteriori error estimator for a quadratic finite element method for the elliptic obstacle problem, Comput. Methods Appl. Math. 15 (2015), no. 2, 145–160.

  • [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 (2009), no. 11, 2774–2784.

    • Crossref
    • Export Citation
  • [25]

    J. Gwinner, hp-FEM convergence for unilateral contact problems with Tresca friction in plane linear elastostatics, J. Comput. Appl. Math. 254 (2013), 175–184.

    • Crossref
    • Export Citation
  • [26]

    P. Hild and S. Nicaise, A posteriori error estimations of residual type for Signorini’s problem, Numer. Math. 101 (2005), no. 3, 523–549.

    • Crossref
    • Export Citation
  • [27]

    P. Hild and Y. Renard, An improved a priori error analysis for finite element approximations of Signorini’s problem, SIAM J. Numer. Anal. 50 (2012), no. 5, 2400–2419.

    • Crossref
    • Export Citation
  • [28]

    M. Hintermüller, K. Ito and K. Kunish, The primal-dual active set strategy as a semismooth Newton method, SIAM J. Optim. 13 (2003), 865–888.

  • [29]

    R. H. W. Hoppe and R. Kornhuber, Adaptive multilevel methods for obstacle problems, SIAM J. Numer. Anal. 31 (1994), no. 2, 301–323.

    • Crossref
    • Export Citation
  • [30]

    S. Hüeber and B. I. Wohlmuth, An optimal a priori error estimate for nonlinear multibody contact problems, SIAM J. Numer. Anal. 43 (2005), no. 1, 156–173.

    • Crossref
    • Export Citation
  • [31]

    S. Kesavan, Functional Analysis, Texts Read. Math. 52, Hindustan Book Agency, New Delhi, 2009.

  • [32]

    D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications. Reprint of the 1980 original, Class. Appl. Math. 31, Society for Industrial and Applied Mathematics, Philadelphia, 2000.

  • [33]

    R. H. Nochetto, K. G. Siebert and A. Veeser, Fully localized a posteriori error estimators and barrier sets for contact problems, SIAM J. Numer. Anal. 42 (2005), no. 5, 2118–2135.

    • Crossref
    • Export Citation
  • [34]

    R. H. Nochetto, T. von Petersdorff and C.-S. Zhang, A posteriori error analysis for a class of integral equations and variational inequalities, Numer. Math. 116 (2010), no. 3, 519–552.

    • Crossref
    • Export Citation
  • [35]

    M. Page and D. Praetorius, Convergence of adaptive FEM for some elliptic obstacle problem, Appl. Anal. 92 (2013), no. 3, 595–615.

    • Crossref
    • Export Citation
  • [36]

    A. Schröder, Mixed finite element methods of higher-order for model contact problems, SIAM J. Numer. Anal. 49 (2011), no. 6, 2323–2339.

    • Crossref
    • Export Citation
  • [37]

    L. R. Scott and S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comp. 54 (1990), no. 190, 483–493.

    • Crossref
    • Export Citation
  • [38]

    K. G. Siebert and A. Veeser, A unilaterally constrained quadratic minimization with adaptive finite elements, SIAM J. Optim. 18 (2007), no. 1, 260–289.

    • Crossref
    • Export Citation
  • [39]

    F. T. Suttmeier, Numerical Solution of Variational Inequalities by Adaptive Finite Elements, Vieweg+Teubner, Wiesbaden, 2008.

  • [40]

    A. Veeser, Efficient and reliable a posteriori error estimators for elliptic obstacle problems, SIAM J. Numer. Anal. 39 (2001), no. 1, 146–167.

    • Crossref
    • Export Citation
  • [41]

    F. Wang, W. Han and X.-L. Cheng, Discontinuous Galerkin methods for solving elliptic variational inequalities, SIAM J. Numer. Anal. 48 (2010), no. 2, 708–733.

    • Crossref
    • Export Citation
  • [42]

    F. Wang, W. Han, J. Eichholz and X. Cheng, A posteriori error estimates for discontinuous Galerkin methods of obstacle problems, Nonlinear Anal. Real World Appl. 22 (2015), 664–679.

    • Crossref
    • Export Citation
  • [43]

    L. Wang, On the quadratic finite element approximation to the obstacle problem, Numer. Math. 92 (2002), no. 4, 771–778.

    • Crossref
    • Export Citation
  • [44]

    A. Weiss and B. I. Wohlmuth, A posteriori error estimator and error control for contact problems, Math. Comp. 78 (2009), no. 267, 1237–1267.

    • Crossref
    • Export Citation
  • [45]

    A. Weiss and B. I. Wohlmuth, A posteriori error estimator for obstacle problems, SIAM J. Sci. Comput. 32 (2010), no. 5, 2627–2658.

    • Crossref
    • Export Citation
  • [46]

    Q. Zou, A. Veeser, R. Kornhuber and C. Gräser, Hierarchical error estimates for the energy functional in obstacle problems, Numer. Math. 117 (2011), no. 4, 653–677.

    • Crossref
    • Export Citation
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