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.
[1] Alart, P., Curnier, A.: A mixed formulation for frictional contact problems prone to Newton like solution methods. Comput. Methods Appl. Mech. Eng. 92(3), 353–375 (1991)
[2] Andersson, L.E.: A quasistatic frictional problem with normal compliance. Nonlinear Anal. Theo. Method. Appl. 16(4), 347–369 (1991)
[3] Andersson, L.E.: Existence results for quasistatic contact problems with Coulomb friction. Appl. Math. Opt. 42, 169–202 (2000)
[4] Andersson, L.E., 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. 359(1789), 2519–2539 (2001)
[5] Barboteu, M., Han, W., Sofonea, M.: Numerical analysis of a bilateral frictional contact problem for linearly elastic materials. IMA J. Numer. Anal. 22(3), 407–436 (2002)
[6] Capatina, A.: Existence and uniqueness results. Var. Inequal. Frict. Contact Probl. 3, 31–82 (2014)
[7] Carstensen, C., Köhler, K.: Nonconforming FEM for the obstacle problem. IMA J. Numer. Anal. 37(1), 64–93 (2017)
[8] Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. SIAM, Philadelphia (2002)
[9] Cocou, M.: A variational inequality and applications to quasistatic problems with Coulomb friction. Appl. Anal. 97(8), 1357–1371 (2018)
[10] Cocu, M.: Existence of solutions of Signorini problems with friction. Int. J. Eng. Sci. 22(5), 567–575 (1984)
[11] Cocu, M., Pratt, E., Raous, M.: Formulation and approximation of quasistatic frictional contact. Int. J. Eng. Sci. 34(7), 783–798 (1996)
[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, J.L.: Inequalities in Mechanics and Physics. Springer Science & Business Media, Berlin (2008)
[15] Fernández, J.R., Hild, P.: A priori and a posteriori error analyses in the study of viscoelastic problems. J. Comput. Appl. Math. 225(2), 569–580 (2009)
[16] Glowinski, R.: Lectures on Numerical Methods for Non-linear Variational Problems. Springer Science & Business Media, Berlin (2008)
[17] Han, W., Sofonea, M.: Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity. American Mathematical Society, Providence, R.I. (2002)
[18] Hansbo, P., Larson, M.G.: Discontinuous Galerkin and the Crouzeix-Raviart element: application to elasticity. ESAIM: Math. Modell. Numer. Anal. 37(1), 63–72 (2003)
[19] Kesavan, S.: Topics in functional analysis and applications. Math. Comp. Sim. 32(4), 424 (1990)
[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. 26(8), 811–832 (1988)
[22] Klarbring, A., Mikelić, A., Shillor, M.: On friction problems with normal compliance. Nonlinear Anal. 13(8), 935–955 (1989)
[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 328(12), 1253–1258 (1999)
[24] Rocca, R., Cocu, M.: Existence and approximation of a solution to quasistatic Signorini problem with local friction. Int. J. Eng. Sci. 39(11), 1233–1255 (2001)
[25] Scott, L.R., Zhang, S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54(190), 483–493 (1990)
[26] Shillor, M., Sofonea, M., Telega, J.J.: Models and Analysis of Quasistatic Contact: Variational Methods, vol. 655. Springer Science & Business Media, Berlin (2004)
[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. 126(4), 771–800 (2014)
