Convergence Analysis of Non-conforming Finite Element Method for a Quasi-static Contact Problem

Abstract

We analyze the numerical solution of a non-linear evolutionary variational inequality, which is encountered in the investigation of quasi-static contact problems. The study of this article encompasses both the semi-discrete and fully discrete schemes, where we employ the backward Euler method for time discretization and utilize the lowest order Crouzeix-Raviart non-conforming finite-element method for spatial discretization. By assuming appropriate regularity conditions on the solution, we establish a priori error analysis for these schemes, achieving the optimal convergence order for linear elements. To illustrate the numerical convergence rates, we provide numerical results on a two-dimensional test problem.

Keywords

Finite-element method / A priori error estimates / Variational inequalities / Quasi-static problem / Supremum norm / 65N30 / 65N15

References

[1] Alart P, Curnier A. A mixed formulation for frictional contact problems prone to Newton like solution methods. Comput. Methods Appl. Mech. Eng.. 1991, 92(3): 353-375.

[2] Andersson LE. A quasistatic frictional problem with normal compliance. Nonlinear Anal. Theo. Method. Appl.. 1991, 16(4): 347-369.

[3] Andersson LE. Existence results for quasistatic contact problems with Coulomb friction. Appl. Math. Opt.. 2000, 42: 169-202.

[4] Andersson LE, Klarbring A. A review of the theory of static and quasi-static frictional contact problems in elasticity. Philos. Trans. R. Soc. London. Ser. A: Math. Phys. Eng. Sci.. 2001, 359(1789): 2519-2539.

[5] Barboteu M, Han W, Sofonea M. Numerical analysis of a bilateral frictional contact problem for linearly elastic materials. IMA J. Numer. Anal.. 2002, 22(3): 407-436.

[6] Capatina A. Existence and uniqueness results. Var. Inequal. Frict. Contact Probl.. 2014, 3: 31-82

[7] Carstensen C, Köhler K. Nonconforming FEM for the obstacle problem. IMA J. Numer. Anal.. 2017, 37(1): 64-93.

[8] Ciarlet PG. The Finite Element Method for Elliptic Problems. 2002, Philadelphia, SIAM.

[9] Cocou M. A variational inequality and applications to quasistatic problems with Coulomb friction. Appl. Anal.. 2018, 97(8): 1357-1371.

[10] Cocu M. Existence of solutions of Signorini problems with friction. Int. J. Eng. Sci.. 1984, 22(5): 567-575

[11] Cocu M, Pratt E, Raous M. Formulation and approximation of quasistatic frictional contact. Int. J. Eng. Sci.. 1996, 34(7): 783-798

[12] Djoko, J.K., Ebobisse, F., McBride, A.T., Reddy, B.: A discontinuous Galerkin formulation for classical and gradient plasticity–Part 1: formulation and analysis. Comput. Methods Appl. Mech. Eng. 196(37/38/39/40), 3881–3897 (2007)

[13] Djoko, J.K., Ebobisse, F., McBride, A.T., Reddy, B.: A discontinuous Galerkin formulation for classical and gradient plasticity. Part 2: algorithms and numerical analysis. Comput. Methods Appl. Mech. Eng. 197(1/2/3/4), 1–21 (2007)

[14] Duvaut G, Lions JL. Inequalities in Mechanics and Physics. 2008, Berlin, Springer Science & Business Media

[15] Fernández JR, Hild P. A priori and a posteriori error analyses in the study of viscoelastic problems. J. Comput. Appl. Math.. 2009, 225(2): 569-580.

[16] Glowinski R. Lectures on Numerical Methods for Non-linear Variational Problems. 2008, Berlin, Springer Science & Business Media

[17] Han, W., Sofonea, M.: Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity. American Mathematical Society, Providence, R.I. (2002)

[18] Hansbo P, Larson MG. Discontinuous Galerkin and the Crouzeix-Raviart element: application to elasticity. ESAIM: Math. Modell. Numer. Anal.. 2003, 37(1): 63-72.

[19] Kesavan S. Topics in functional analysis and applications. Math. Comp. Sim.. 1990, 32(4): 424.

[20] Kikuchi, N., Oden, J.T.: Contact Problems in Elasticity: a Study of Variational Inequalities and Finite Element Methods. SIAM, Philadelphia (1988)

[21] Klarbring A, Mikelić A, Shillor M. Frictional contact problems with normal compliance. Int. J. Eng. Sci.. 1988, 268811-832

[22] Klarbring A, Mikelić A, Shillor M. On friction problems with normal compliance. Nonlinear Anal.. 1989, 13(8): 935-955.

[23] Rocca R. Existence of a solution for a quasistatic problem of unilateral contact with local friction. Comptes Rendus de l’Académie des Sciences-Series I-Mathematics. 1999, 328(12): 1253-1258

[24] Rocca R, Cocu M. Existence and approximation of a solution to quasistatic Signorini problem with local friction. Int. J. Eng. Sci.. 2001, 39(11): 1233-1255

[25] Scott LR, Zhang S. Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput.. 1990, 54(190): 483-493.

[26] Shillor M, Sofonea M, Telega JJ. Models and Analysis of Quasistatic Contact: Variational Methods. 2004, Berlin, Springer Science & Business Media655

[27] Sofonea, M., Matei, A.: Variational inequalities with applications: a study of antiplane frictional contact problems, In: Advances in Mechanics and Mathematics, vol. 18. Springer Science & Business Media, Berlin (2009)

[28] Wang F, Han W, Cheng X. Discontinuous Galerkin methods for solving a quasistatic contact problem. Numer. Math.. 2014, 126(4): 771-800.

RIGHTS & PERMISSIONS

Shanghai University