doi: 10.3934/era.2020103

On $ P_1 $ 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  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 $.

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 BelgacemP. 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 BelgacemP. 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. BrezziW. 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. LiQ. 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 BelgacemP. 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 BelgacemP. 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. BrezziW. 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. LiQ. 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
$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
$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
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