Discontinuous Galerkin methods for a contact problem with Tresca friction arising in linear elasticity. (English) Zbl 1381.74207
Summary: In this article, we propose and analyze discontinuous Galerkin (DG) methods for a contact problem with Tresca friction for the linearized elastic material. We derive a residual based a posteriori error estimator for the proposed class of DG methods. The reliability and the efficiency of a posteriori error estimator is shown. We further investigate a priori error estimates under the minimal regularity assumption on the exact solution. An important property shared by a class of DG methods, allow us to carry out the analysis in a unified framework. Numerical experiments are reported to illustrate theoretical results.
MSC:
| 74S05 | Finite element methods applied to problems in solid mechanics |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65K15 | Numerical methods for variational inequalities and related problems |
| 65N15 | Error bounds for boundary value problems involving PDEs |
| 74B05 | Classical linear elasticity |
| 35Q74 | PDEs in connection with mechanics of deformable solids |
Keywords:
finite element; discontinuous Galerkin; a posteriori error estimate; frictional contact problem; variational inequalities; linear elasticityReferences:
| [1] | Andres, M., Dual-Dual Formulations for Frictional Contact Problems in Mechanics (2011), Leibniz Universität Hannover, PhD thesis |
| [2] | Arnold, D. N., An interior penalty finite element method with discontinuous elements, SIAM J. Numer. Anal., 19, 742-760 (1982) · Zbl 0482.65060 |
| [3] | Arnold, D. N.; Brezzi, F.; Cockburn, B.; Marini, L. D., Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal., 39, 1749-1779 (2002) · Zbl 1008.65080 |
| [4] | Atkinson, K.; Han, W., Theoretical Numerical Analysis. A Functional Analysis Framework (2009), Springer · Zbl 1181.47078 |
| [5] | Bartels, S.; Carstensen, C., Averaging techniques yield reliable a posteriori finite element error control for obstacle problems, Numer. Math., 99, 225-249 (2004) · Zbl 1063.65050 |
| [6] | Bassi, F.; Rebay, S.; Mariotti, G.; Pedinotti, S.; Savini, M., A high order accurate discontinuous finite element method for inviscid and viscous turbomachinery flows, (Decuypere, R.; Dibelius, G., Proceedings of 2nd European Conference on Turbomachinery, Fluid Dynamics and Thermodynamics (1997), Technologisch Instituut: Technologisch Instituut Antwerpen, Belgium), 99-178 |
| [7] | Blum, H.; Suttmeier, F. T., An adaptive finite element discretization for a simplified Signorini problem, Calcolo, 37, 65-77 (2000) · Zbl 0954.65053 |
| [8] | Bostan, V.; Han, W., Recovery-based error estimation and adaptive solution of elliptic variational inequalities of the second kind, Commun. Math. Sci., 2, 1-18 (2004) · Zbl 1084.35025 |
| [9] | Bostan, V.; Han, W., A posteriori error analysis for finite element solutions of a frictional contact problem, Comput. Methods Appl. Mech. Eng., 195, 1252-1274 (2006) · Zbl 1115.74047 |
| [10] | Bostan, V.; Han, W., Adaptive finite element solution of variational inequalities with application in contact problem, (Gao, D. Y.; Sherali, H. D., Advances in Applied Mathematics and Global Optimization (2009), Springer: Springer New York), 25-106 · Zbl 1193.49006 |
| [11] | Bostan, V.; Han, W.; Reddy, B., A posteriori error estimation and adaptive solution of elliptic variational inequalities of the second kind, Appl. Numer. Math., 52, 13-38 (2004) · Zbl 1069.65067 |
| [12] | Braess, D., A posteriori error estimators for obstacle problems-another look, Numer. Math., 101, 415-421 (2005) · Zbl 1118.65068 |
| [13] | Brenner, S., Two-level additive Schwarz preconditioners for nonconforming finite element methods, Math. Comput., 65, 897-921 (1996) · Zbl 0859.65124 |
| [14] | Brenner, S., Convergence of nonconforming multigrid methods without full elliptic regularity, Math. Comput., 68, 25-53 (1999) · Zbl 0912.65099 |
| [15] | Brenner, S., Ponicaré-Friedrichs inequalities for piecewise \(H^1\) functions, SIAM J. Numer. Anal., 41, 306-324 (2003) · Zbl 1045.65100 |
| [16] | Brenner, S., Korn’s inequalities for piecewise \(H^1\) vector fields, Math. Comput., 73, 1067-1087 (2004) · Zbl 1055.65118 |
| [17] | Brenner, S.; Scott, L., The Mathematical Theory of Finite Element Methods (2008), Springer-Verlag: Springer-Verlag New York · Zbl 1135.65042 |
| [18] | Brenner, S.; Sung, L. Y.; Zhang, Y., Finite element methods for the displacement obstacle problem of clamped plates, Math. Comput., 81, 1247-1262 (2012) · Zbl 1250.74023 |
| [19] | Brezzi, F.; Hager, W. W.; Raviart, P. A., Error estimates for the finite element solution of variational inequalities, Part I. Primal theory, Numer. Math., 28, 431-443 (1977) · Zbl 0369.65030 |
| [20] | Brezzi, F.; Manzini, G.; Marini, D.; Pietra, P.; Russo, A., Discontinuous finite elements for diffusion problems, (Atti Convegno in onore di F. Brioschi. Atti Convegno in onore di F. Brioschi, Milan, 1997 (1999), Istituto Lombardo, Accademia di Scienze e Lettere: Istituto Lombardo, Accademia di Scienze e Lettere Milan, Italy), 197-217 |
| [21] | Brezzi, F.; Manzini, G.; Marini, D.; Pietra, P.; Russo, A., Discontinuous Galerkin approximations for elliptic problems, Numer. Methods Partial Differ. Equ., 16, 365-378 (2000) · Zbl 0957.65099 |
| [22] | Bürg, M.; Schröder, A., A posteriori error control of hp-finite elements for variational inequalities of the first and second kind, Comput. Math. Appl., 70, 2783-2802 (2015) · Zbl 1443.65314 |
| [23] | Bustinza, R.; Sayas, F. J., Error estimates for an LDG method applied to a Signorini type problems, J. Sci. Comput., 52, 322-339 (2012) · Zbl 1311.74110 |
| [24] | Castillo, P.; Cockburn, B.; Perugia, I.; Schötzau, D., An a priori error analysis of the local discontinuous Galerkin method for elliptic problems, SIAM J. Numer. Anal., 38, 1676-1706 (2000) · Zbl 0987.65111 |
| [25] | Chen, Z.; Nochetto, R., Residual type a posteriori error estimates for elliptic obstacle problems, Numer. Math., 84, 527-548 (2000) · Zbl 0943.65075 |
| [26] | Ciarlet, P., The Finite Element Method for Elliptic Problems (1978), North-Holland: North-Holland Amsterdam · Zbl 0383.65058 |
| [27] | Cockburn, B.; Shu, C. W., The local discontinuous Galerkin method for time-dependent convection-diffusion systems, SIAM J. Numer. Anal., 35, 2440-2463 (1998) · Zbl 0927.65118 |
| [28] | Dörlfer, W., A convergent adaptive algorithm for Poisson’s equation, SIAM J. Numer. Anal., 33, 1106-1124 (1996) · Zbl 0854.65090 |
| [29] | Dörsek, P.; Melenk, J. M., Adaptive hp-fem for the contact problem with Tresca friction in linear elasticity: the primal-dual formulation and a posteriori error estimation, Appl. Numer. Math., 60, 689-704 (2010) · Zbl 1306.74036 |
| [30] | Duvaut, G.; Lions, J., Inequalities in Mechanics and Physics (1976), Springer: Springer Berlin · Zbl 0331.35002 |
| [31] | Falk, R. S., Error estimates for the approximation of a class of variational inequalities, Math. Comput., 28, 963-971 (1974) · Zbl 0297.65061 |
| [32] | Glowinski, R., Numerical Methods for Nonlinear Variational Problems (1984), Springer-Verlag: Springer-Verlag New York · Zbl 0575.65123 |
| [33] | Gudi, T., A new error analysis for discontinuous finite element methods for linear elliptic problems, Math. Comput., 79, 2169-2189 (2010) · Zbl 1201.65198 |
| [34] | Gudi, T.; Porwal, K., A posteriori error control of discontinuous Galerkin methods for elliptic obstacle problems, Math. Comput., 83, 579-602 (2014) · Zbl 1305.65231 |
| [35] | Gudi, T.; Porwal, K., A remark on the a posteriori error analysis of discontinuous Galerkin methods for obstacle problem, Comput. Methods Appl. Math., 14, 71-87 (2014) · Zbl 1285.65041 |
| [36] | Gudi, T.; Porwal, K., An a posteriori error estimator for a class of discontinuous Galerkin methods for Signorini problem, J. Comput. Appl. Math., 292, 257-278 (2016) · Zbl 1327.65239 |
| [37] | Gudi, T.; Porwal, K., A C0 interior penalty method for a fourth-order variational inequality of the second kind, Numer. Methods Partial Differ. Equ., 32, 36-59 (2016) · Zbl 1339.65088 |
| [38] | Hage, D.; Klein, N.; Suttmeier, F., Adaptive finite elements for a certain class of variational inequalities of the second kind, Calcolo, 48, 293-305 (2011) · Zbl 1239.65042 |
| [39] | Han, W.; Wang, L., Nonconforming finite element analysis for a plate contact problem, SIAM J. Numer. Anal., 40, 1683-1697 (2002) · Zbl 1034.74045 |
| [40] | Hild, P.; Nicaise, S., A posteriori error estimates of residual type for Signorini’s problem, Numer. Math., 101, 523-549 (2005) · Zbl 1084.65106 |
| [41] | Hild, P.; Nicaise, S., Residual a posteriori error estimators for contact problems in elasticity, ESAIM: Math. Model. Numer. Anal., 41, 897-923 (2007) · Zbl 1140.74024 |
| [42] | Douglas, J. J.; Dupont, T., Interior Penalty Procedures for Elliptic and Parabolic Galerkin Methods, Lect. Notes Phys., vol. 58 (1976), Springer-Verlag: Springer-Verlag Berlin |
| [43] | Karakashian, O.; Pascal, F., A posteriori error estimates for a discontinuous Galerkin approximation of second-order elliptic problems, SIAM J. Numer. Anal., 41, 2374-2399 (2003) · Zbl 1058.65120 |
| [44] | Kikuchi, N.; Oden, J. T., Contact Problems in Elasticity (1988), SIAM: SIAM Philadelphia · Zbl 0685.73002 |
| [45] | Kinderlehrer, D.; Stampacchia, G., An Introduction to Variational Inequalities and Their Applications (2000), SIAM: SIAM Philadelphia · Zbl 0988.49003 |
| [46] | Nochetto, R.; Petersdorff, T. V.; Zhang, C. S., A posteriori error analysis for a class of integral equations and variational inequalities, Numer. Math., 116, 519-552 (2010) · Zbl 1202.65085 |
| [47] | Panagiotopoulos, P., Inequality Problems in Mechanics and Applications (1985), Birkhäuser: Birkhäuser Boston · Zbl 0579.73014 |
| [48] | Rademacher, A.; Schröder, A., Dual weighted residual error control for frictional contact problems, Comput. Methods Appl. Math., 15, 391-413 (2015) · Zbl 1317.65239 |
| [49] | Reed, W. H.; Hill, T. R., Triangular Mesh Methods for the Neutron Transport Equation (1973), Los Alamos Scientific Laboratory, Technical report LA-UR-73-479 |
| [50] | Rivière, B.; Wheeler, M.; Girault, V., A priori error estimates for finite element methods based on discontinuous approximation spaces for elliptic problems, SIAM J. Numer. Anal., 39, 902-931 (2001) · Zbl 1010.65045 |
| [51] | Schröder, A., Mixed finite element methods of higher-order for model contact problems, SIAM J. Numer. Anal., 49, 2323-2339 (2011) · Zbl 1298.74236 |
| [52] | Suttmeier, F. T., General approach for a posteriori error estimates for finite element solutions of variational inequalities, Comput. Mech., 27, 317-323 (2001) · Zbl 0987.65057 |
| [53] | Veeser, A., Efficient and reliable a posteriori error estimators for elliptic obstacle problems, SIAM J. Numer. Anal., 39, 146-167 (2001) · Zbl 0992.65073 |
| [54] | Wang, F.; Han, W.; Cheng, X., Discontinuous Galerkin methods for solving elliptic variational inequalities, SIAM J. Numer. Anal., 48, 708-733 (2010) · Zbl 1214.65039 |
| [55] | Wang, F.; Han, W.; Cheng, X., Discontinuous Galerkin methods for solving the Signorini problem, IMA J. Numer. Anal., 41, 1754-1772 (2011) · Zbl 1315.74021 |
| [56] | Wang, F.; Han, W.; Cheng, X., Another view for a posteriori error estimates for variational inequalities of the second kind, Appl. Numer. Math., 78, 225-233 (2013) · Zbl 1302.65159 |
| [57] | Wang, L., On the quadratic finite element approximation to the obstacle problem, Numer. Math., 92, 771-778 (2002) · Zbl 1049.65062 |
| [58] | Weiss, A.; Wohlmuth, B. I., A posteriori error estimator and error control for contact problems, Math. Comput., 78, 1237-1267 (2009) · Zbl 1198.65234 |
| [59] | Wheeler, M., An elliptic collocation-finite-element method with interior penalties, SIAM J. Numer. Anal., 15, 152-161 (1978) · Zbl 0384.65058 |
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