Pointwise A Posteriori Error Control of Discontinuous Galerkin Methods for Unilateral Contact Problems

References

[1] M. Ainsworth and J. T. Oden, A Posteriori Error Estimation in Finite Element Analysis, John Wiley & Sons, New York, 2011. Search in Google Scholar

[2] L. Ambrosio, Lecture notes on elliptic partial differential equations, unpublished lecture notes, Scuola Normale Superiore di Pisa, 2015. Search in Google Scholar

[3] C. Baiocchi, Estimations d’erreur dans L ∞ pour les inéquations à obstacle, Mathematical Aspects of Finite Element Methods (Rome 1975), Lecture Notes in Math. 606, Springer, Berlin (1977), 27–34. 10.1007/BFb0064453Search in Google Scholar

[4] 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), no. 4, 1198–1216. 10.1137/S0036142998347966Search in Google Scholar

[5] S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Texts Appl. Math. 15, Springer, New York, 2007. 10.1007/978-0-387-75934-0Search in Google Scholar

[6] F. Brezzi, W. W. Hager and P.-A. Raviart, Error estimates for the finite element solution of variational inequalities, Numer. Math. 28 (1977), no. 4, 431–443. 10.1007/BF01404345Search in Google Scholar

[7] R. Bustinza and F.-J. Sayas, Error estimates for an LDG method applied to Signorini type problems, J. Sci. Comput. 52 (2012), no. 2, 322–339. 10.1007/s10915-011-9548-5Search in Google Scholar

[8] L. A. Caffarelli, Further regularity for the Signorini problem, Comm. Partial Differential Equations 4 (1979), no. 9, 1067–1075. 10.1080/03605307908820119Search in Google Scholar

[9] P. G. Ciarlet, The Finite Element Method for Elliptic Problems, Classics Appl. Math. 40, Society for Industrial and Applied Mathematics, Philadelphia, 2002. 10.1137/1.9780898719208Search in Google Scholar

[10] A. Demlow and E. H. Georgoulis, Pointwise a posteriori error control for discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal. 50 (2012), no. 5, 2159–2181. 10.1137/110846397Search in Google Scholar

[11] G. Dolzmann and S. Müller, Estimates for Green’s matrices of elliptic systems by L p theory, Manuscripta Math. 88 (1995), no. 2, 261–273. 10.1007/BF02567822Search in Google Scholar

[12] F. Fierro and A. Veeser, A posteriori error estimators for regularized total variation of characteristic functions, SIAM J. Numer. Anal. 41 (2003), no. 6, 2032–2055. 10.1137/S0036142902408283Search in Google Scholar

[13] A. Friedman, Variational Principles and Free-Boundary Problems, John Wiley & Sons, New York, 1982. Search in Google Scholar

[14] R. Glowinski, Lectures on Numerical Methods for Nonlinear Variational Problems, Tata Institute of Fundamental Research, Bombay, 1980. Search in Google Scholar

[15] T. Gudi and K. Porwal, A posteriori error estimates of discontinuous Galerkin methods for the Signorini problem, J. Comput. Appl. Math. 292 (2016), 257–278. 10.1016/j.cam.2015.07.008Search in Google Scholar

[16] S. Hofmann and S. Kim, The Green function estimates for strongly elliptic systems of second order, Manuscripta Math. 124 (2007), no. 2, 139–172. 10.1007/s00229-007-0107-1Search in Google Scholar

[17] S. Hüeber and B. I. Wohlmuth, A primal-dual active set strategy for non-linear multibody contact problems, Comput. Methods Appl. Mech. Engrg. 194 (2005), no. 27–29, 3147–3166. 10.1016/j.cma.2004.08.006Search in Google Scholar

[18] O. A. Karakashian and F. Pascal, A posteriori error estimates for a discontinuous Galerkin approximation of second-order elliptic problems, SIAM J. Numer. Anal. 41 (2003), no. 6, 2374–2399. 10.1137/S0036142902405217Search in Google Scholar

[19] T. Kashiwabara and T. Kemmochi, Pointwise error estimates of linear finite element method for Neumann boundary value problems in a smooth domain, Numer. Math. 144 (2020), no. 3, 553–584. 10.1007/s00211-019-01098-8Search in Google Scholar

[20] S. Kesavan, Topics in Functional Analysis and Applications, John Wiley & Sons, New York, 1989. Search in Google Scholar

[21] N. Kikuchi and J. T. Oden, Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods, Society for Industrial and Applied Mathematics, Philadelphia, 1988. 10.1137/1.9781611970845Search in Google Scholar

[22] D. Kinderlehrer, Remarks about Signorini’s problem in linear elasticity, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 8 (1981), no. 4, 605–645. Search in Google Scholar

[23] R. Krause, A. Veeser and M. Walloth, An efficient and reliable residual-type a posteriori error estimator for the Signorini problem, Numer. Math. 130 (2015), no. 1, 151–197. 10.1007/s00211-014-0655-8Search in Google Scholar

[24] M. Li, D. Hua and H. Lian, On P 1 nonconforming finite element aproximation for the Signorini problem, Electron. Res. Arch. 29 (2021), no. 2, 2029–2045. 10.3934/era.2020103Search in Google Scholar

[25] K.-S. Moon, R. H. Nochetto, T. von Petersdorff and C.-S. Zhang, A posteriori error analysis for parabolic variational inequalities, M2AN Math. Model. Numer. Anal. 41 (2007), no. 3, 485–511. 10.1051/m2an:2007029Search in Google Scholar

[26] J. Nitsche, L ∞ -convergence of finite element approximations, Mathematical Aspects of Finite Element Methods (Rome 1975), Lecture Notes in Math. 606, Springer, Berlin (1977), 261–274 10.1007/BFb0064468Search in Google Scholar

[27] R. H. Nochetto, Pointwise a posteriori error estimates for elliptic problems on highly graded meshes, Math. Comp. 64 (1995), no. 209, 1–22. 10.1090/S0025-5718-1995-1270622-3Search in Google Scholar

[28] R. H. Nochetto, M. Paolini and C. Verdi, An adaptive finite element method for two-phase Stefan problems in two space dimensions. II. Implementation and numerical experiments, SIAM J. Sci. Statist. Comput. 12 (1991), no. 5, 1207–1244. 10.1137/0912065Search in Google Scholar

[29] R. H. Nochetto, A. Schmidt, K. G. Siebert and A. Veeser, Pointwise a posteriori error estimates for monotone semi-linear equations, Numer. Math. 104 (2006), no. 4, 515–538. 10.1007/s00211-006-0027-0Search in Google Scholar

[30] R. H. Nochetto, K. G. Siebert and A. Veeser, Pointwise a posteriori error control for elliptic obstacle problems, Numer. Math. 95 (2003), no. 1, 163–195. 10.1007/s00211-002-0411-3Search in Google Scholar

[31] R. H. Nochetto, K. G. Siebert and A. Veeser, Fully localized a posteriori error estimators and barrier sets for contact problems, SIAM J. Numer. Anal. 42 (2005), no. 5, 2118–2135. 10.1137/S0036142903424404Search in Google Scholar

[32] F. Scarpini and M. A. Vivaldi, Error estimates for the approximation of some unilateral problems, RAIRO Anal. Numér. 11 (1977), no. 2, 197–208. 10.1051/m2an/1977110201971Search in Google Scholar

[33] J. L. Taylor, S. Kim and R. M. Brown, The Green function for elliptic systems in two dimensions, Comm. Partial Differential Equations 38 (2013), no. 9, 1574–1600. 10.1080/03605302.2013.814668Search in Google Scholar

[34] R. Verfürth, A Review of a Posteriori Error Estimation and Adaptive Mesh-refinement Techniques, Wiley, Chichester, 1996. Search in Google Scholar

[35] M. Walloth, Adaptive numerical simulation of contact problems: Resolving local effects at the contact boundary in space and time, Ph.D. thesis, Rheinische Friedrich-Wilhelms-Universität Bonn, 2012. Search in Google Scholar

[36] M. Walloth, A reliable, efficient and localized error estimator for a discontinuous Galerkin method for the Signorini problem, Appl. Numer. Math. 135 (2019), 276–296. 10.1016/j.apnum.2018.09.002Search in Google Scholar

[37] F. Wang, W. Han and X. Cheng, Discontinuous Galerkin methods for solving the Signorini problem, IMA J. Numer. Anal. 31 (2011), no. 4, 1754–1772. 10.1093/imanum/drr010Search in Google Scholar

[38] A. Weiss and B. I. Wohlmuth, A posteriori error estimator and error control for contact problems, Math. Comp. 78 (2009), no. 267, 1237–1267. 10.1090/S0025-5718-09-02235-2Search in Google Scholar