### Electronic Research Archive

2021, Issue 2: 2029-2045. doi: 10.3934/era.2020103
Special Issues

# On $P_1$ nonconforming finite element aproximation for the Signorini problem

• Received: 01 May 2020 Revised: 01 July 2020 Published: 23 September 2020
• 65N30, 35J86

• The main aim of this paper is to study the $P_1$ 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 $H^{2}(\Omega)$ without any assumption. Furthermore, we also study the optimal convergence for the case $u\in H^{1+\nu}(\Omega)$ with $\frac{1}{2}<\nu<1$.

Citation: Mingxia Li, Dongying Hua, Hairong Lian. On $P_1$ nonconforming finite element aproximation for the Signorini problem[J]. Electronic Research Archive, 2021, 29(2): 2029-2045. doi: 10.3934/era.2020103

### Related Papers:

• The main aim of this paper is to study the $P_1$ 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 $H^{2}(\Omega)$ without any assumption. Furthermore, we also study the optimal convergence for the case $u\in H^{1+\nu}(\Omega)$ with $\frac{1}{2}<\nu<1$.

 [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] $P_1$ 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] R$\acute{e}$gularit$\acute{e}$ des solutions d'un probl$\grave{e}$m m$\hat{e}$l$\acute{e}$ 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 $2$nd 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.
###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142

1.833 0.8

Article outline

Tables(2)

• On This Site