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Computational Methods in Applied Mathematics

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Higher Order Mixed FEM for the Obstacle Problem of the p-Laplace Equation Using Biorthogonal Systems

Lothar Banz / Bishnu P. Lamichhane
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  • School of Mathematical & Physical Sciences, University of Newcastle, University Drive, Callaghan, NSW 2308, Australia
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/ Ernst P. Stephan
Published Online: 2018-06-21 | DOI: https://doi.org/10.1515/cmam-2018-0015

Abstract

We consider a mixed finite element method for an obstacle problem with the p-Laplace differential operator for p(1,), where the obstacle condition is imposed by using a Lagrange multiplier. In the discrete setting the Lagrange multiplier basis forms a biorthogonal system with the standard finite element basis so that the variational inequality can be realized in the point-wise form. We provide a general a posteriori error estimate for adaptivity and prove an a priori error estimate. We present numerical results for the adaptive scheme (mesh-size adaptivity with and without polynomial degree adaptation) for the singular case p=1.5 and the degenerated case p=3. We also present numerical results on the mesh independency and on the polynomial degree scaling of the discrete inf-sup constant when using biorthogonal basis functions for the dual variable defined on the same mesh with the same polynomial degree distribution.

Keywords: A Priori Error Estimate; A Posteriori Error Estimate; Discrete Inf-Sup Constant

MSC 2010: 65N30; 65N15; 74M15

References

  • [1]

    L. Banz, B. P. Lamichhane and E. P. Stephan, Higher order FEM for the obstacle problem of the p-Laplacian – A variational inequality approach, preprint, (2017).

  • [2]

    L. Banz and A. Schröder, Biorthogonal basis functions in hp-adaptive FEM for elliptic obstacle problems, Comput. Math. Appl. 70 (2015), no. 8, 1721–1742. CrossrefWeb of ScienceGoogle Scholar

  • [3]

    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. CrossrefWeb of ScienceGoogle Scholar

  • [4]

    L. Banz and E. P. Stephan, hp-adaptive IPDG/TDG-FEM for parabolic obstacle problems, Comput. Math. Appl. 67 (2014), 712–731. Web of ScienceCrossrefGoogle Scholar

  • [5]

    J. W. Barrett and W. Liu, Quasi-norm error bounds for the finite element approximation of a non-Newtonian flow, Numer. Math. 68 (1994), no. 4, 437–456. CrossrefGoogle Scholar

  • [6]

    S. Bartels and C. Carstensen, Averaging techniques yield reliable a posteriori finite element error control for obstacle problems, Numer. Math. 99 (2004), 225–249. CrossrefGoogle Scholar

  • [7]

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

  • [8]

    D. Braess, C. Carstensen and R. Hoppe, Convergence analysis of a conforming adaptive finite element method for an obstacle problem, Numer. Math. 107 (2007), 455–471. CrossrefWeb of ScienceGoogle Scholar

  • [9]

    S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, 3 ed., Springer, New York, 2008. Google Scholar

  • [10]

    C. Carstensen and J. Hu, An optimal adaptive finite element method for an obstacle problem, Comput. Methods Appl. Math. 15 (2015), 259–277. Web of ScienceGoogle Scholar

  • [11]

    C. Carstensen and R. Klose, A posteriori finite element error control for the p-Laplace problem, SIAM J. Sci. Comput. 25 (2003), no. 3, 792–814. CrossrefGoogle Scholar

  • [12]

    C. Carstensen, W. Liu and N. Yan, A posteriori FE error control for p-Laplacian by gradient recovery in quasi-norm, Math. Comp. 75 (2006), no. 256, 1599–1616. CrossrefGoogle Scholar

  • [13]

    M. Chipot, Elliptic Equations: An Introductory Course, Birkhäuser, Basel, 2009. Google Scholar

  • [14]

    L. Diening and C. Kreuzer, Linear convergence of an adaptive finite element method for the p-Laplacian equation, SIAM J. Numer. Anal. 46 (2008), no. 2, 614–638. Web of ScienceCrossrefGoogle Scholar

  • [15]

    C. Ebmeyer and W. Liu, Quasi-norm interpolation error estimates for the piecewise linear finite element approximation of p-Laplacian problems, Numer. Math. 100 (2005), no. 2, 233–258. CrossrefGoogle Scholar

  • [16]

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

  • [17]

    S. Hüeber and B. Wohlmuth, A primal–dual active set strategy for non-linear multibody contact problems, Comput. Methods Appl. Mech. Engrg. 194 (2005), no. 27–29, 3147–3166. CrossrefGoogle Scholar

  • [18]

    G. Jouvet and E. Bueler, Steady, shallow ice sheets as obstacle problems: Well-posedness and finite element approximation, SIAM J. Appl. Math. 72 (2012), no. 4, 1292–1314. Web of ScienceCrossrefGoogle Scholar

  • [19]

    R. Krause, B. Müller and G. Starke, An adaptive least-squares mixed finite element method for the signorini problem, Numer. Methods Partial Differential Equations 33 (2017), 276–289. CrossrefWeb of ScienceGoogle Scholar

  • [20]

    A. Krebs and E. Stephan, A p-version finite element method for nonlinear elliptic variational inequalities in 2D, Numer. Math. 105 (2007), no. 3, 457–480. Web of ScienceGoogle Scholar

  • [21]

    B. Lamichhane and B. Wohlmuth, Biorthogonal bases with local support and approximation properties, Math. Comp. 76 (2007), no. 257, 233–249. CrossrefGoogle Scholar

  • [22]

    M. Lewicka and J. J. Manfredi, The obstacle problem for the p-Laplacian via optimal stopping of tug-of-war games, Probab. Theory Related Fields 167 (2017), no. 1–2, 349–378. CrossrefWeb of ScienceGoogle Scholar

  • [23]

    W. Liu and N. Yan, Quasi-norm local error estimators for p-Laplacian, SIAM J. Numer. Anal. 39 (2001), no. 1, 100–127. CrossrefGoogle Scholar

  • [24]

    W. Liu and N. Yan, On quasi-norm interpolation error estimation and a posteriori error estimates for p-Laplacian, SIAM J. Numer. Anal. 40 (2002), no. 5, 1870–1895. CrossrefGoogle Scholar

  • [25]

    M. Maischak and E. P. Stephan, Adaptive hp-versions of BEM for Signorini problems, Appl. Numer. Math. 54 (2005), no. 3, 425–449. CrossrefGoogle Scholar

  • [26]

    D. Malkus, Eigenproblems associated with the discrete LBB condition for incompressible finite elements, Internat. J. Engrg. Sci. 19 (1981), no. 10, 1299–1310. CrossrefGoogle Scholar

  • [27]

    J. M. Melenk, hp-interpolation of nonsmooth functions and an application to hp-a posteriori error estimation, SIAM J. Numer. Anal. 43 (2005), no. 1, 127–155. CrossrefGoogle Scholar

  • [28]

    N. Ovcharova and L. Banz, Coupling regularization and adaptive hp-BEM for the solution of a delamination problem, Numer. Math. 137 (2017), 303–337. Web of ScienceCrossrefGoogle Scholar

  • [29]

    J. Qin, On the convergence of some low order mixed finite elements for incompressible fluids, Ph.D. thesis, The Pennsylvania State University, 1994. Google Scholar

  • [30]

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

  • [31]

    T. Wick, An error-oriented Newton/inexact augmented Lagrangian approach for fully monolithic phase-field fracture propagation, SIAM J. Sci. Comput. 39 (2017), no. 4, B589–B617. Web of ScienceGoogle Scholar

About the article

Received: 2017-12-12

Revised: 2018-04-30

Accepted: 2018-06-07

Published Online: 2018-06-21


The visit of the third author to the University of Newcastle, Australia was partially supported by the priority research centre of the University of Newcastle for Computer-Assisted Research Mathematics and its Applications.


Citation Information: Computational Methods in Applied Mathematics, ISSN (Online) 1609-9389, ISSN (Print) 1609-4840, DOI: https://doi.org/10.1515/cmam-2018-0015.

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