American Institute of Mathematical Sciences

Advanced
doi: 10.3934/era.2020103

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

Mingxia Li 1, , Dongying Hua 2, and  Hairong Lian 1,

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  May 2020 Revised  July 2020 Published  September 2020

Fund Project: This paper is supported by the China Scholarship Council (No. 201806405013) and China Major Science and Technology Program for Water Pollution Control and Treatment (2018ZX07109-002)

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 $.

Keywords: Variational inequality, Signorini problem, nonconforming finite element, optimal error estimate.
Mathematics Subject Classification: 65N30, 35J86.
Citation: Mingxia Li, Dongying Hua, Hairong Lian. On $ P_1 $ nonconforming finite element aproximation for the Signorini problem. Electronic Research Archive, doi: 10.3934/era.2020103
References:
[1]

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), 1198-1216.  doi: 10.1137/S0036142998347966.  Google Scholar

[2]

F. Ben Belgacem and Y. Renard, Hybrid finite element methods for the Signorini problem, Math. Comp., 72 (2003), 1117-1145.  doi: 10.1090/S0025-5718-03-01490-X.  Google Scholar

[3]

F. Ben Belgacem, P. Hild and P. Laborde, Extension of the motar finite element to a variational inequality modeling unilateral contact, Math. Models. Methods Appl. Sci., 9 (1999), 287-303.  doi: 10.1142/S0218202599000154.  Google Scholar

[4]

F. Ben Belgacem, P. Hild and P. Laborde, Approximation of the unilateral contact problem by the motor finite element method, C. R. Acad. Sci. Paris Sér. I. Math., 324 (1997), 123-127.  doi: 10.1016/S0764-4442(97)80115-2.  Google Scholar

[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.  Google Scholar

[6]

S. C. Brenner and L.-Y. Sung, Linear finite element methods for planar elasticity, Math. Comp., 59 (1992), 321-338.  doi: 10.1090/S0025-5718-1992-1140646-2.  Google Scholar

[7]

F. Brezzi, W. W. Hager and P.-A. Raviart, Error estimates for the finite element solution of variational inequalities, Numer. Math., 28 (1977), 431-443.  doi: 10.1007/BF01404345.  Google Scholar

[8]

D. Capatina-Papaghiuc and J.-M. Thomas, Nonconforming finite element methods without numerical locking, Numer. Math., 81 (1998), 163-186.  doi: 10.1007/s002110050388.  Google Scholar

[9]

M. Crouzeix and P.-A. Raviart, Conforming and Nonconforming finite element methods for solving the stationary Stokes problems. I., Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge, 7 (1973), 33-76.   Google Scholar

[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.  Google Scholar

[11]

R. S. Falk, Nonconforming finite element methods for the equations of linear elasticity, Math. Comp., 57 (1991), 529-550.  doi: 10.1090/S0025-5718-1991-1094947-6.  Google Scholar

[12]

P. Hild, Nonconforming finite elements for unilateral contact with friction, C. R. Acad. Sci. Paris Sér. I. Math., 324 (1997), 707-710.  doi: 10.1016/S0764-4442(97)86994-7.  Google Scholar

[13]

D. Y. Hua and L. H. Wang, $P_1$ Nonconforming Finite Element Approximation of Unilateral Problem, J. Comp. Math., 25 (2007), 67-80.   Google Scholar

[14]

S. H$\ddot{u}$eber and B. I. Wohlmuth, An optimal a priori error estimate for nonlinear multibody contact problems, SIAM J. Numer. Math., 43 (2005), 156-173.  doi: 10.1137/S0036142903436678.  Google Scholar

[15]

R. Kouhia and R. Stenberg, A linear nonconforming finite element method for nearly incompressible elasticity and Stokes flow, Comput. Methods Appl. Mech. Engrg., 124 (1995), 195-212.  doi: 10.1016/0045-7825(95)00829-P.  Google Scholar

[16]

K. Lhalouani and T. Sassi, Nonconforming mixed variational formulation and domain decomposition for unilateral problems, East-West J. Numer. Math., 7 (1999), 23-30.   Google Scholar

[17]

M.-X. Li, Q. Lin and S.-H. Zhang, Superconvergence of finite element method for the Signorini problem, J. Comput. Appl. Math., 222 (2008), 284-292.  doi: 10.1016/j.cam.2007.10.058.  Google Scholar

[18]

M. Moussaoui and K. Khodja, 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., 17 (1992), 805-826.  doi: 10.1080/03605309208820864.  Google Scholar

[19]

R. Rannacher and S. Turek, Simple nonconforming quadrilateral Stokes element, Numer. Meth. PDE., 8 (1992), 97-111.  doi: 10.1002/num.1690080202.  Google Scholar

[20]

P.-A. Raviart and J. M. Thomas, Primal hybrid finite element methods for $2$nd order elliptic equations, Math. Comp., 31 (1977), 391-413.  doi: 10.2307/2006423.  Google Scholar

[21]

F. Scarpini and M. A. Vivaldi, Error estimates for the approximation of some unilateral problems, RAIRO Anal. Numér., 11 (1977), 197-208.  doi: 10.1051/m2an/1977110201971.  Google Scholar

[22]

L. Slimane and Y. Renard, The treat of the locking phenomenon for a general class of variational inequalities, J. Comp. Appl. Math., 170 (2004), 121-143.  doi: 10.1016/j.cam.2003.12.044.  Google Scholar

[23]

L.-H. Wang, Nonconforming finite element approximation of unilateral problem, J. Comp. Math., 17 (1999), 15-24.   Google Scholar

show all references

References:
[1]

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), 1198-1216.  doi: 10.1137/S0036142998347966.  Google Scholar

[2]

F. Ben Belgacem and Y. Renard, Hybrid finite element methods for the Signorini problem, Math. Comp., 72 (2003), 1117-1145.  doi: 10.1090/S0025-5718-03-01490-X.  Google Scholar

[3]

F. Ben Belgacem, P. Hild and P. Laborde, Extension of the motar finite element to a variational inequality modeling unilateral contact, Math. Models. Methods Appl. Sci., 9 (1999), 287-303.  doi: 10.1142/S0218202599000154.  Google Scholar

[4]

F. Ben Belgacem, P. Hild and P. Laborde, Approximation of the unilateral contact problem by the motor finite element method, C. R. Acad. Sci. Paris Sér. I. Math., 324 (1997), 123-127.  doi: 10.1016/S0764-4442(97)80115-2.  Google Scholar

[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.  Google Scholar

[6]

S. C. Brenner and L.-Y. Sung, Linear finite element methods for planar elasticity, Math. Comp., 59 (1992), 321-338.  doi: 10.1090/S0025-5718-1992-1140646-2.  Google Scholar

[7]

F. Brezzi, W. W. Hager and P.-A. Raviart, Error estimates for the finite element solution of variational inequalities, Numer. Math., 28 (1977), 431-443.  doi: 10.1007/BF01404345.  Google Scholar

[8]

D. Capatina-Papaghiuc and J.-M. Thomas, Nonconforming finite element methods without numerical locking, Numer. Math., 81 (1998), 163-186.  doi: 10.1007/s002110050388.  Google Scholar

[9]

M. Crouzeix and P.-A. Raviart, Conforming and Nonconforming finite element methods for solving the stationary Stokes problems. I., Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge, 7 (1973), 33-76.   Google Scholar

[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.  Google Scholar

[11]

R. S. Falk, Nonconforming finite element methods for the equations of linear elasticity, Math. Comp., 57 (1991), 529-550.  doi: 10.1090/S0025-5718-1991-1094947-6.  Google Scholar

[12]

P. Hild, Nonconforming finite elements for unilateral contact with friction, C. R. Acad. Sci. Paris Sér. I. Math., 324 (1997), 707-710.  doi: 10.1016/S0764-4442(97)86994-7.  Google Scholar

[13]

D. Y. Hua and L. H. Wang, $P_1$ Nonconforming Finite Element Approximation of Unilateral Problem, J. Comp. Math., 25 (2007), 67-80.   Google Scholar

[14]

S. H$\ddot{u}$eber and B. I. Wohlmuth, An optimal a priori error estimate for nonlinear multibody contact problems, SIAM J. Numer. Math., 43 (2005), 156-173.  doi: 10.1137/S0036142903436678.  Google Scholar

[15]

R. Kouhia and R. Stenberg, A linear nonconforming finite element method for nearly incompressible elasticity and Stokes flow, Comput. Methods Appl. Mech. Engrg., 124 (1995), 195-212.  doi: 10.1016/0045-7825(95)00829-P.  Google Scholar

[16]

K. Lhalouani and T. Sassi, Nonconforming mixed variational formulation and domain decomposition for unilateral problems, East-West J. Numer. Math., 7 (1999), 23-30.   Google Scholar

[17]

M.-X. Li, Q. Lin and S.-H. Zhang, Superconvergence of finite element method for the Signorini problem, J. Comput. Appl. Math., 222 (2008), 284-292.  doi: 10.1016/j.cam.2007.10.058.  Google Scholar

[18]

M. Moussaoui and K. Khodja, 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., 17 (1992), 805-826.  doi: 10.1080/03605309208820864.  Google Scholar

[19]

R. Rannacher and S. Turek, Simple nonconforming quadrilateral Stokes element, Numer. Meth. PDE., 8 (1992), 97-111.  doi: 10.1002/num.1690080202.  Google Scholar

[20]

P.-A. Raviart and J. M. Thomas, Primal hybrid finite element methods for $2$nd order elliptic equations, Math. Comp., 31 (1977), 391-413.  doi: 10.2307/2006423.  Google Scholar

[21]

F. Scarpini and M. A. Vivaldi, Error estimates for the approximation of some unilateral problems, RAIRO Anal. Numér., 11 (1977), 197-208.  doi: 10.1051/m2an/1977110201971.  Google Scholar

[22]

L. Slimane and Y. Renard, The treat of the locking phenomenon for a general class of variational inequalities, J. Comp. Appl. Math., 170 (2004), 121-143.  doi: 10.1016/j.cam.2003.12.044.  Google Scholar

[23]

L.-H. Wang, Nonconforming finite element approximation of unilateral problem, J. Comp. Math., 17 (1999), 15-24.   Google Scholar

Table 5.1.  $L^2$-error and norm error based on $u_{256}$
$h$$\|u_{Ch}-u_{256}\|_{0, \Omega}$$\|u_{NCh}-u_{256}\|_{0, \Omega}$$\|u_{Ch}-u_{256}\|_{h}$$\|u_{NCh}-u_{256}\|_{h}$
$1/2$0.100757110.141821680.690496840.72541038
$1/4$0.028227910.023183350.345696870.31868546
$1/8$0.008584960.006374380.181554190.16303402
$1/16$0.001999930.001634250.092694700.08283960
$1/32$0.000534860.000424390.046583880.04170061
$1/64$0.000143560.000098350.023359610.02090943
a.c.r.1.891002462.098773000.977109341.02331438
Table Options

Download as excel

$h$$\|u_{Ch}-u_{256}\|_{0, \Omega}$$\|u_{NCh}-u_{256}\|_{0, \Omega}$$\|u_{Ch}-u_{256}\|_{h}$$\|u_{NCh}-u_{256}\|_{h}$
$1/2$0.100757110.141821680.690496840.72541038
$1/4$0.028227910.023183350.345696870.31868546
$1/8$0.008584960.006374380.181554190.16303402
$1/16$0.001999930.001634250.092694700.08283960
$1/32$0.000534860.000424390.046583880.04170061
$1/64$0.000143560.000098350.023359610.02090943
a.c.r.1.891002462.098773000.977109341.02331438
Table 5.2.  $L^2$-error and norm error based on $u_{512}$
$h$$\|u_{Ch}-u_{512}\|_{0, \Omega}$$\|u_{NCh}-u_{512}\|_{0, \Omega}$$\|u_{Ch}-u_{512}\|_{h}$$\|u_{NCh}-u_{512}\|_{h}$
$1/2$0.100754950.141821500.690487760.72540780
$1/4$0.028225810.023183980.345683110.31868764
$1/8$0.008582780.006374500.181529950.16304308
$1/16$0.001998020.001634320.092648090.08286176
$1/32$0.000532740.000424400.046491830.04174806
$1/64$0.000141340.000098660.023175790.02100774
a.c.r.1.895492942.097864720.979385091.02195988
Table Options

Download as excel

$h$$\|u_{Ch}-u_{512}\|_{0, \Omega}$$\|u_{NCh}-u_{512}\|_{0, \Omega}$$\|u_{Ch}-u_{512}\|_{h}$$\|u_{NCh}-u_{512}\|_{h}$
$1/2$0.100754950.141821500.690487760.72540780
$1/4$0.028225810.023183980.345683110.31868764
$1/8$0.008582780.006374500.181529950.16304308
$1/16$0.001998020.001634320.092648090.08286176
$1/32$0.000532740.000424400.046491830.04174806
$1/64$0.000141340.000098660.023175790.02100774
a.c.r.1.895492942.097864720.979385091.02195988
[1]

Liupeng Wang, Yunqing Huang. Error estimates for second-order SAV finite element method to phase field crystal model. Electronic Research Archive, 2021, 29 (1) : 1735-1752. doi: 10.3934/era.2020089
[2]

Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020340
[3]

Ying Liu, Yanping Chen, Yunqing Huang, Yang Wang. Two-grid method for semiconductor device problem by mixed finite element method and characteristics finite element method. Electronic Research Archive, 2021, 29 (1) : 1859-1880. doi: 10.3934/era.2020095
[4]

Gang Bao, Mingming Zhang, Bin Hu, Peijun Li. An adaptive finite element DtN method for the three-dimensional acoustic scattering problem. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 61-79. doi: 10.3934/dcdsb.2020351
[5]

Zuliang Lu, Fei Huang, Xiankui Wu, Lin Li, Shang Liu. Convergence and quasi-optimality of $ L^2- $norms based an adaptive finite element method for nonlinear optimal control problems. Electronic Research Archive, 2020, 28 (4) : 1459-1486. doi: 10.3934/era.2020077
[6]

P. K. Jha, R. Lipton. Finite element approximation of nonlocal dynamic fracture models. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1675-1710. doi: 10.3934/dcdsb.2020178
[7]

Constantine M. Dafermos. A variational approach to the Riemann problem for hyperbolic conservation laws. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 185-195. doi: 10.3934/dcds.2009.23.185
[8]

Yue Feng, Yujie Liu, Ruishu Wang, Shangyou Zhang. A conforming discontinuous Galerkin finite element method on rectangular partitions. Electronic Research Archive, , () : -. doi: 10.3934/era.2020120
[9]

Bin Wang, Lin Mu. Viscosity robust weak Galerkin finite element methods for Stokes problems. Electronic Research Archive, 2021, 29 (1) : 1881-1895. doi: 10.3934/era.2020096
[10]

Xiu Ye, Shangyou Zhang, Peng Zhu. A weak Galerkin finite element method for nonlinear conservation laws. Electronic Research Archive, 2021, 29 (1) : 1897-1923. doi: 10.3934/era.2020097
[11]

Jiwei Jia, Young-Ju Lee, Yue Feng, Zichan Wang, Zhongshu Zhao. Hybridized weak Galerkin finite element methods for Brinkman equations. Electronic Research Archive, , () : -. doi: 10.3934/era.2020126
[12]

Shumin Li, Masahiro Yamamoto, Bernadette Miara. A Carleman estimate for the linear shallow shell equation and an inverse source problem. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 367-380. doi: 10.3934/dcds.2009.23.367
[13]

Wenya Qi, Padmanabhan Seshaiyer, Junping Wang. A four-field mixed finite element method for Biot's consolidation problems. Electronic Research Archive, , () : -. doi: 10.3934/era.2020127
[14]

Lingfeng Li, Shousheng Luo, Xue-Cheng Tai, Jiang Yang. A new variational approach based on level-set function for convex hull problem with outliers. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020070
[15]

Xiaoli Lu, Pengzhan Huang, Yinnian He. Fully discrete finite element approximation of the 2D/3D unsteady incompressible magnetohydrodynamic-Voigt regularization flows. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 815-845. doi: 10.3934/dcdsb.2020143
[16]

Xin Zhang, Jie Xiong, Shuaiqi Zhang. Optimal reinsurance-investment and dividends problem with fixed transaction costs. Journal of Industrial & Management Optimization, 2021, 17 (2) : 981-999. doi: 10.3934/jimo.2020008
[17]

Yifan Chen, Thomas Y. Hou. Function approximation via the subsampled Poincaré inequality. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 169-199. doi: 10.3934/dcds.2020296
[18]

Andrew D. Lewis. Erratum for "nonholonomic and constrained variational mechanics". Journal of Geometric Mechanics, 2020, 12 (4) : 671-675. doi: 10.3934/jgm.2020033
[19]

Alberto Bressan, Wen Shen. A posteriori error estimates for self-similar solutions to the Euler equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 113-130. doi: 10.3934/dcds.2020168
[20]

Xing-Bin Pan. Variational and operator methods for Maxwell-Stokes system. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3909-3955. doi: 10.3934/dcds.2020036

 Impact Factor: 0.263

Tools

Article outline

Figures and Tables

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences