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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 |
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 $.
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. |
[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. |
[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. |
[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. |
[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] |
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. |
[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. |
[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. |
[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.
|
[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] |
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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[19] |
R. Rannacher and S. Turek,
Simple nonconforming quadrilateral Stokes element, Numer. Meth. PDE., 8 (1992), 97-111.
doi: 10.1002/num.1690080202. |
[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. |
[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. |
[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. |
[23] |
L.-H. Wang,
Nonconforming finite element approximation of unilateral problem, J. Comp. Math., 17 (1999), 15-24.
|
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. |
[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. |
[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. |
[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. |
[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] |
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. |
[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. |
[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. |
[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.
|
[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] |
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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[19] |
R. Rannacher and S. Turek,
Simple nonconforming quadrilateral Stokes element, Numer. Meth. PDE., 8 (1992), 97-111.
doi: 10.1002/num.1690080202. |
[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. |
[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. |
[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. |
[23] |
L.-H. Wang,
Nonconforming finite element approximation of unilateral problem, J. Comp. Math., 17 (1999), 15-24.
|
$h$ | $\|u_{Ch}-u_{256}\|_{0, \Omega}$ | $\|u_{NCh}-u_{256}\|_{0, \Omega}$ | $\|u_{Ch}-u_{256}\|_{h}$ | $\|u_{NCh}-u_{256}\|_{h}$ |
$1/2$ | 0.10075711 | 0.14182168 | 0.69049684 | 0.72541038 |
$1/4$ | 0.02822791 | 0.02318335 | 0.34569687 | 0.31868546 |
$1/8$ | 0.00858496 | 0.00637438 | 0.18155419 | 0.16303402 |
$1/16$ | 0.00199993 | 0.00163425 | 0.09269470 | 0.08283960 |
$1/32$ | 0.00053486 | 0.00042439 | 0.04658388 | 0.04170061 |
$1/64$ | 0.00014356 | 0.00009835 | 0.02335961 | 0.02090943 |
a.c.r. | 1.89100246 | 2.09877300 | 0.97710934 | 1.02331438 |
$h$ | $\|u_{Ch}-u_{256}\|_{0, \Omega}$ | $\|u_{NCh}-u_{256}\|_{0, \Omega}$ | $\|u_{Ch}-u_{256}\|_{h}$ | $\|u_{NCh}-u_{256}\|_{h}$ |
$1/2$ | 0.10075711 | 0.14182168 | 0.69049684 | 0.72541038 |
$1/4$ | 0.02822791 | 0.02318335 | 0.34569687 | 0.31868546 |
$1/8$ | 0.00858496 | 0.00637438 | 0.18155419 | 0.16303402 |
$1/16$ | 0.00199993 | 0.00163425 | 0.09269470 | 0.08283960 |
$1/32$ | 0.00053486 | 0.00042439 | 0.04658388 | 0.04170061 |
$1/64$ | 0.00014356 | 0.00009835 | 0.02335961 | 0.02090943 |
a.c.r. | 1.89100246 | 2.09877300 | 0.97710934 | 1.02331438 |
$h$ | $\|u_{Ch}-u_{512}\|_{0, \Omega}$ | $\|u_{NCh}-u_{512}\|_{0, \Omega}$ | $\|u_{Ch}-u_{512}\|_{h}$ | $\|u_{NCh}-u_{512}\|_{h}$ |
$1/2$ | 0.10075495 | 0.14182150 | 0.69048776 | 0.72540780 |
$1/4$ | 0.02822581 | 0.02318398 | 0.34568311 | 0.31868764 |
$1/8$ | 0.00858278 | 0.00637450 | 0.18152995 | 0.16304308 |
$1/16$ | 0.00199802 | 0.00163432 | 0.09264809 | 0.08286176 |
$1/32$ | 0.00053274 | 0.00042440 | 0.04649183 | 0.04174806 |
$1/64$ | 0.00014134 | 0.00009866 | 0.02317579 | 0.02100774 |
a.c.r. | 1.89549294 | 2.09786472 | 0.97938509 | 1.02195988 |
$h$ | $\|u_{Ch}-u_{512}\|_{0, \Omega}$ | $\|u_{NCh}-u_{512}\|_{0, \Omega}$ | $\|u_{Ch}-u_{512}\|_{h}$ | $\|u_{NCh}-u_{512}\|_{h}$ |
$1/2$ | 0.10075495 | 0.14182150 | 0.69048776 | 0.72540780 |
$1/4$ | 0.02822581 | 0.02318398 | 0.34568311 | 0.31868764 |
$1/8$ | 0.00858278 | 0.00637450 | 0.18152995 | 0.16304308 |
$1/16$ | 0.00199802 | 0.00163432 | 0.09264809 | 0.08286176 |
$1/32$ | 0.00053274 | 0.00042440 | 0.04649183 | 0.04174806 |
$1/64$ | 0.00014134 | 0.00009866 | 0.02317579 | 0.02100774 |
a.c.r. | 1.89549294 | 2.09786472 | 0.97938509 | 1.02195988 |
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