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Crouzeix–Raviart Finite Element Approximation for the Parabolic Obstacle Problem

  • Thirupathi Gudi ORCID logo EMAIL logo and Papri Majumder

Abstract

We introduce and study a fully discrete nonconforming finite element approximation for a parabolic variational inequality associated with a general obstacle problem. The method comprises of the Crouzeix–Raviart finite element method for space discretization and implicit backward Euler scheme for time discretization. We derive an error estimate of optimal order 𝒪(h+Δt) in a certain energy norm defined precisely in the article. We only assume the realistic regularity utL2(0,T;L2(Ω)) and moreover the analysis is performed without any assumptions on the speed of propagation of the free boundary. We present a numerical experiment to illustrate the theoretical order of convergence derived in the article.

MSC 2010: 65N30; 65N15

References

[1] J. Alberty, C. Carstensen and S. A. Funken, Remarks around 50 lines of Matlab: Short finite element implementation, Numer. Algorithms 20 (1999), no. 2–3, 117–137. 10.1023/A:1019155918070Search in Google Scholar

[2] C. Bahriawati and C. Carstensen, Three MATLAB implementations of the lowest-order Raviart–Thomas MFEM with a posteriori error control, Comput. Methods Appl. Math. 5 (2005), no. 4, 333–361. 10.2478/cmam-2005-0016Search in Google Scholar

[3] L. Banz and E. P. Stephan, hp-adaptive IPDG/TDG-FEM for parabolic obstacle problems, Comput. Math. Appl. 67 (2014), no. 4, 712–731. 10.1016/j.camwa.2013.03.003Search in Google Scholar

[4] A. E. Berger and R. S. Falk, An error estimate for the truncation method for the solution of parabolic obstacle variational inequalities, Math. Comp. 31 (1977), no. 139, 619–628. 10.1090/S0025-5718-1977-0438707-8Search in Google Scholar

[5] S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, 3rd ed., Texts Appl. Math. 15, Springer, New York, 2008. 10.1007/978-0-387-75934-0Search in Google Scholar

[6] H. Brézis, Problèmes unilatéraux, J. Math. Pures Appl. (9) 51 (1972), 1–168. Search in Google Scholar

[7] H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland Math. Stud. 5, North-Holland, Amsterdam, 1973. Search in Google Scholar

[8] C. Carstensen, J. Gedicke and D. Rim, Explicit error estimates for Courant, Crouzeix–Raviart and Raviart–Thomas finite element methods, J. Comput. Math. 30 (2012), no. 4, 337–353. 10.4208/jcm.1108-m3677Search in Google Scholar

[9] C. Carstensen and K. Köhler, Nonconforming FEM for the obstacle problem, IMA J. Numer. Anal. 37 (2017), no. 1, 64–93. 10.1093/imanum/drw005Search in Google Scholar

[10] Z. Chen and R. H. Nochetto, Residual type a posteriori error estimates for elliptic obstacle problems, Numer. Math. 84 (2000), no. 4, 527–548. 10.1007/s002110050009Search in Google Scholar

[11] P. G. Ciarlet, The Finite Element Method for Elliptic Problems, Stud. Math. Appl. 4, North-Holland, Amsterdam, 1978. 10.1115/1.3424474Search in Google Scholar

[12] M. Crouzeix and P.-A. Raviart, Conforming and nonconforming finite element methods for solving the stationary Stokes equations. I, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 7 (1973), no. R-3, 33–75. 10.1051/m2an/197307R300331Search in Google Scholar

[13] S. Gaddam and T. Gudi, Inhomogeneous Dirichlet boundary condition in the a posteriori error control of the obstacle problem, Comput. Math. Appl. 75 (2018), no. 7, 2311–2327. 10.1016/j.camwa.2017.12.010Search in Google Scholar

[14] R. Glowinski, J. L. Lions and R. Thémoliéres, Numerical Methods for Variational Inequalities, North-Holland, Amsterdam, 1981. Search in Google Scholar

[15] T. Gudi and P. Majumder, Conforming and discontinuous Galerkin FEM in space for solving parabolic obstacle problem, Comput. Math. Appl. (2019), 10.1016/j.camwa.2019.06.022. 10.1016/j.camwa.2019.06.022Search in Google Scholar

[16] T. Gudi and P. Majumder, Convergence analysis of finite element method for a parabolic obstacle problem, J. Comput. Appl. Math. 357 (2019), 85–102. 10.1016/j.cam.2019.02.026Search in Google Scholar

[17] A. Fetter, L-error estimate for an approximation of a parabolic variational inequality, Numer. Math. 50 (1987), no. 5, 557–565. 10.1007/BF01408576Search in Google Scholar

[18] M. Hintermüller, K. Ito and K. Kunisch, The primal-dual active set strategy as a semismooth Newton method, SIAM J. Optim. 13 (2002), no. 3, 865–888. 10.1137/S1052623401383558Search in Google Scholar

[19] M. A. B. L. Hocine, S. Boulaaras and M. Haiour, An optimal L-error estimate for an approximation of a parabolic variational inequality, Numer. Funct. Anal. Optim. 37 (2016), no. 1, 1–18. 10.1080/01630563.2015.1109520Search in Google Scholar

[20] C. Johnson, A convergence estimate for an approximation of a parabolic variational inequality, SIAM J. Numer. Anal. 13 (1976), no. 4, 599–606. 10.1137/0713050Search in Google Scholar

[21] S. Kesavan, Functional Analysis, Texts Read. Math. 52, Hindustan Book, New Delhi, 2009. 10.1007/978-93-86279-42-2Search in Google Scholar

[22] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Class. Appl. Math. 31, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 2000. 10.1137/1.9780898719451Search in Google Scholar

[23] J. L. Lions, Partial differential inequalities, Uspehi Mat. Nauk 26 (1971), no. 2(158), 205–263; translation in Russian Math. Surveys 27, no. 2, 91–159. 10.1070/RM1972v027n02ABEH001373Search in Google Scholar

[24] J.-L. Lions and G. Stampacchia, Variational inequalities, Comm. Pure Appl. Math. 20 (1967), 493–519. 10.1002/cpa.3160200302Search 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] R. H. Nochetto, G. Savaré and C. Verdi, Error control of nonlinear evolution equations, C. R. Acad. Sci. Paris Sér. I Math. 326 (1998), no. 12, 1437–1442. 10.1016/S0764-4442(98)80407-2Search in Google Scholar

[27] R. H. Nochetto, G. Savaré and C. Verdi, A posteriori error estimates for variable time-step discretizations of nonlinear evolution equations, Comm. Pure Appl. Math. 53 (2000), no. 5, 525–589. 10.1002/(SICI)1097-0312(200005)53:5<525::AID-CPA1>3.0.CO;2-MSearch in Google Scholar

[28] E. Otárola and A. J. Salgado, Finite element approximation of the parabolic fractional obstacle problem, SIAM J. Numer. Anal. 54 (2016), no. 4, 2619–2639. 10.1137/15M1029801Search in Google Scholar

[29] A. K. Pani and P. C. Das, A priori error estimates for a single-phase quasilinear Stefan problem in one space dimension, IMA J. Numer. Anal. 11 (1991), no. 3, 377–392. 10.1093/imanum/11.3.377Search in Google Scholar

[30] J. Rulla, Error analysis for implicit approximations to solutions to Cauchy problems, SIAM J. Numer. Anal. 33 (1996), no. 1, 68–87. 10.1137/0733005Search in Google Scholar

[31] G. Savaré, Weak solutions and maximal regularity for abstract evolution inequalities, Adv. Math. Sci. Appl. 6 (1996), no. 2, 377–418. Search in Google Scholar

[32] G. Stampacchia, Èquations elliptiques du second ordre à coefficients discontinus, Sémin. Math. Supér. 16, Les Presses de l’Université de Montréal, Montreal, 1966. Search in Google Scholar

[33] V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, 2nd ed., Springer Ser. Comput. Math. 25, Springer, Berlin, 2006. Search in Google Scholar

[34] C. Vuik, An L2-error estimate for an approximation of the solution of a parabolic variational inequality, Numer. Math. 57 (1990), no. 5, 453–471. 10.1007/BF01386423Search in Google Scholar

[35] X. Yang, G. Wang and X. Gu, Numerical solution for a parabolic obstacle problem with nonsmooth initial data, Numer. Methods Partial Differential Equations 30 (2014), no. 5, 1740–1754. 10.1002/num.21893Search in Google Scholar

[36] C.-S. Zhang, Adaptive Finite Element Methods for Variational Inequalities: Theory and Applications in Finance, ProQuest LLC, Ann Arbor, 2007; Thesis (Ph.D.)–University of Maryland, College Park. Search in Google Scholar

Received: 2018-09-13
Revised: 2019-08-05
Accepted: 2019-08-06
Published Online: 2019-08-28
Published in Print: 2020-04-01

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