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2021, Volume 29, Issue 2: 2029-2045. doi: 10.3934/era.2020103
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On P1 nonconforming finite element aproximation for the Signorini problem

  • 1.
    School of Science, China University of Geosciences, Beijing 100083, China
  • 2.
    School of Applied Science, Beijing Information Science and Technology University, Beijing 100192, China
  • Received: 01 May 2020 Revised: 01 July 2020 Published: 23 September 2020

  • 65N30, 35J86

  • The main aim of this paper is to study the P1 nonconforming finite element approximations of the variational inequality arisen from the Signorini problem. We describe the finite dimensional closed convex cone approximation in a meanvalue-oriented sense. In this way, the optimal convergence rate O(h) can be obtained by a refined analysis when the exact solution belongs to H2(Ω) without any assumption. Furthermore, we also study the optimal convergence for the case u∈H1+ν(Ω) with 12<ν<1.

    Citation: Mingxia Li, Dongying Hua, Hairong Lian. On P1 nonconforming finite element aproximation for the Signorini problem[J]. Electronic Research Archive, 2021, 29(2): 2029-2045. doi: 10.3934/era.2020103

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  • The main aim of this paper is to study the P1 nonconforming finite element approximations of the variational inequality arisen from the Signorini problem. We describe the finite dimensional closed convex cone approximation in a meanvalue-oriented sense. In this way, the optimal convergence rate O(h) can be obtained by a refined analysis when the exact solution belongs to H2(Ω) without any assumption. Furthermore, we also study the optimal convergence for the case u∈H1+ν(Ω) with 12<ν<1.





    [1] Numerical simulation of some variational inequalities arisen from unilateral contact problems by the finite element methods. SIAM J. Numer. Anal. (2000) 37: 1198-1216.
    [2] Hybrid finite element methods for the Signorini problem. Math. Comp. (2003) 72: 1117-1145.
    [3] Extension of the motar finite element to a variational inequality modeling unilateral contact. Math. Models. Methods Appl. Sci. (1999) 9: 287-303.
    [4] Approximation of the unilateral contact problem by the motor finite element method. C. R. Acad. Sci. Paris Sér. I. Math. (1997) 324: 123-127.
    [5] S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4757-4338-8
    [6] Linear finite element methods for planar elasticity. Math. Comp. (1992) 59: 321-338.
    [7] Error estimates for the finite element solution of variational inequalities. Numer. Math. (1977) 28: 431-443.
    [8] Nonconforming finite element methods without numerical locking. Numer. Math. (1998) 81: 163-186.
    [9] Conforming and Nonconforming finite element methods for solving the stationary Stokes problems. I.. Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge (1973) 7: 33-76.
    [10] O. Dorok, V. John, U. Risch, F. Schieweck and L. Tobiska, Parallel finite element methods for the incompressible Navier-Stokes equations, In: Flow Simulation with High-Performance Computers II (E. H. Hirschel ed.). Notes on Numerical Fluid Mechanics, 52 (1996), 20–33.
    [11] Nonconforming finite element methods for the equations of linear elasticity. Math. Comp. (1991) 57: 529-550.
    [12] Nonconforming finite elements for unilateral contact with friction. C. R. Acad. Sci. Paris Sér. I. Math. (1997) 324: 707-710.
    [13] P1 Nonconforming Finite Element Approximation of Unilateral Problem. J. Comp. Math. (2007) 25: 67-80.
    [14] An optimal a priori error estimate for nonlinear multibody contact problems. SIAM J. Numer. Math. (2005) 43: 156-173.
    [15] A linear nonconforming finite element method for nearly incompressible elasticity and Stokes flow. Comput. Methods Appl. Mech. Engrg. (1995) 124: 195-212.
    [16] Nonconforming mixed variational formulation and domain decomposition for unilateral problems. East-West J. Numer. Math. (1999) 7: 23-30.
    [17] Superconvergence of finite element method for the Signorini problem. J. Comput. Appl. Math. (2008) 222: 284-292.
    [18] Rˊegularitˊe des solutions d'un problˊem mˆelˊe Dirichlet-Signorini dans un domaine polygonal plan. Comm. Part. Diff. Eq. (1992) 17: 805-826.
    [19] Simple nonconforming quadrilateral Stokes element. Numer. Meth. PDE. (1992) 8: 97-111.
    [20] Primal hybrid finite element methods for 2nd order elliptic equations. Math. Comp. (1977) 31: 391-413.
    [21] Error estimates for the approximation of some unilateral problems. RAIRO Anal. Numér. (1977) 11: 197-208.
    [22] The treat of the locking phenomenon for a general class of variational inequalities. J. Comp. Appl. Math. (2004) 170: 121-143.
    [23] Nonconforming finite element approximation of unilateral problem. J. Comp. Math. (1999) 17: 15-24.

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