Applications of Mathematics, Vol. 66, No. 5, pp. 673-699, 2021


Unified error analysis of discontinuous Galerkin methods for parabolic obstacle problem

Papri Majumder

Received April 20, 2020.   Published online May 25, 2021.

Abstract:  We introduce and study various discontinuous Galerkin (DG) finite element approximations for a parabolic variational inequality associated with a general obstacle problem in $\mathbb{R}^d$ $(d=2,3)$. For the fully-discrete DG scheme, we employ a piecewise linear finite element space for spatial discretization, whereas the time discretization is carried out with the implicit backward Euler method. We present a unified error analysis for all well known symmetric and non-symmetric DG fully discrete schemes, and derive error estimate of optimal order $\mathcal{O}(h+\Delta t)$ in an energy norm. Moreover, the analysis is performed without any assumptions on the speed of propagation of the free boundary and only the realistic regularity $u_t\in\mathcal{L}^2(0,T; \mathcal{L}^2(\Omega))$ is assumed. Further, we present some numerical experiments to illustrate the performance of the proposed methods.
Keywords:  finite element; discontinuous Galerkin method; parabolic obstacle problem
Classification MSC:  65N30, 65N15


References:
[1] D. N. Arnold: An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19 (1982), 742-760. DOI 10.1137/0719052 | MR 0664882 | Zbl 0482.65060
[2] D. N. Arnold, F. Brezzi, B. Cockburn, L. D. Marini: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39 (2002), 1749-1779. DOI 10.1137/S0036142901384162 | MR 1885715 | Zbl 1008.65080
[3] I. Babuška, M. Zlámal: Nonconforming elements in the finite element method with penalty. SIAM J. Numer. Anal. 10 (1973), 863-875. DOI 10.1137/0710071 | MR 0345432 | Zbl 0237.65066
[4] L. Banz, E. P. Stephan: $hp$-adaptive IPDG/TDG-FEM for parabolic obstacle problems. Comput. Math. Appl. 67 (2014), 712-731. DOI 10.1016/j.camwa.2013.03.003 | MR 3163875 | Zbl 1350.65064
[5] F. Bassi, S. Rebay, G. Mariotti, S. Pedinotti, M. Savini: A high order accurate discontinuous finite element method for inviscid and viscous turbomachinery flows. Proceedings of 2nd European Conference on Turbomachinery, Fluid Dynamics and Thermodynamics (R. Decuypere, G. Dibelius, eds.). Technologisch Instituut, Antwerpen (1997), 99-108.
[6] A. E. Berger, R. S. Falk: An error estimate for the truncation method for the solution of parabolic obstacle variational inequalities. Math. Comput. 31 (1977), 619-628. DOI 10.1090/S0025-5718-1977-0438707-8 | MR 0438707 | Zbl 0367.65056
[7] S. C. Brenner, L. Owens, L.-Y. Sung: A weakly over-penalized symmetric interior penalty method. ETNA, Electron. Tran. Numer. Anal. 30 (2008), 107-127. MR 2480072 | Zbl 1171.65077
[8] H. Brézis: Problèmes unilatéraux. J. Math. Pures Appl. (9) 51 (1972), 1-168. (In French.) MR 0428137 | Zbl 0237.35001
[9] H. Brézis: Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert. North-Holland Mathematics Studies 5. North-Holland, Amsterdam (1973). (In French.) DOI 10.1016/s0304-0208(08)x7125-7 | MR 0348562 | Zbl 0252.47055
[10] F. Brezzi, G. Manzini, D. Marini, P. Pietra, A. Russo: Discontinuous finite elements for diffusion problems. Francesco Brioschi (1824-1897) Convegno di Studi Matematici. Istituto Lombardo, Accademia di Scienze e Lettere, Milan (1999), 197-217.
[11] F. Brezzi, G. Manzini, D. Marini, P. Pietra, A. Russo: Discontinuous Galerkin approximations for elliptic problems. Numer. Methods Partial Differ. Equations 16 (2000), 365-378. DOI 10.1002/1098-2426(200007)16:4<365::AID-NUM2>3.0.CO;2-Y | MR 1765651 | Zbl 0957.65099
[12] P. Castillo, B. Cockburn, I. Perugia, D. Schötzau: An a priori error analysis of the local discontinuous Galerkin method for elliptic problems. SIAM J. Numer. Anal. 38 (2000), 1676-1706. DOI 10.1137/S0036142900371003 | MR 1813251 | Zbl 0987.65111
[13] J. Česenek, M. Feistauer: Theory of the space-time discontinuous Galerkin method for nonstationary parabolic problems with nonlinear convection and diffusion. SIAM J. Numer. Anal. 50 (2012), 1181-1206. DOI 10.1137/110828903 | MR 2970739 | Zbl 1312.65157
[14] Z. Chen, R. H. Nochetto: Residual type a posteriori error estimates for elliptic obstacle problems. Numer. Math. 84 (2000), 527-548. DOI 10.1007/s002110050009 | MR 1742264 | Zbl 0943.65075
[15] P. G. Ciarlet: The Finite Element Method for Elliptic Problems. Studies in Mathematics and Its Applications 4. North-Holland, Amsterdam (1978). DOI 10.1137/1.9780898719208 | MR 0520174 | Zbl 0383.65058
[16] B. Cockburn, G. Kanschat, I. Perugia, D. Schötzau: Superconvergence of the local discontinuous Galerkin method for elliptic problems on Cartesian grids. SIAM J. Numer. Anal. 39 (2001), 264-285. DOI 10.1137/S0036142900371544 | MR 1860725 | Zbl 1041.65080
[17] B. Cockburn, C.-W. Shu: The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J. Numer. Anal. 35 (1998), 2440-2463. DOI 10.1137/S0036142997316712 | MR 1655854 | Zbl 0927.65118
[18] A. Fetter: $L^\infty$-error estimate for an approximation of a parabolic variational inequality. Numer. Math. 50 (1987), 557-565. DOI 10.1007/BF01408576 | MR 0880335 | Zbl 0617.65064
[19] V. Girault, B. Riviére, M. F. Wheeler: A discontinuous Galerkin method with nonoverlapping domain decomposition for the Stokes and Navier-Stokes problems. Math. Comput. 74 (2005), 53-84. DOI 10.1090/S0025-5718-04-01652-7 | MR 2085402 | Zbl 1057.35029
[20] R. Glowinski, J.-L. Lions, R. Trémolières: Numerical Methods for Variational Inequalities. Studies in Mathematics and Its Applications 8. North-Holland, Amsterdam (1981). DOI 10.1016/s0168-2024(08)x7016-x | MR 0635927 | Zbl 0463.65046
[21] T. Gudi, P. Majumder: Conforming and discontinuous Galerkin FEM in space for solving parabolic obstacle problem. Comput. Math. Appl. 78 (2019), 3896-3915. DOI 10.1016/j.camwa.2019.06.022 | MR 4029105 | Zbl 1443.65203
[22] T. Gudi, P. Majumder: Convergence analysis of finite element method for a parabolic obstacle problem. J. Comput. Appl. Math. 357 (2019), 85-102. DOI 10.1016/j.cam.2019.02.026 | MR 3922211 | Zbl 1418.65173
[23] T. Gudi, P. Majumder: Crouzeix-Raviart finite element approximation for the parabolic obstacle problem. Comput. Methods Appl. Math. 20 (2020), 273-292. DOI 10.1515/cmam-2019-0057 | MR 4080232 | Zbl 1436.65182
[24] T. Gudi, N. Nataraj, A. K. Pani: $hp$-discontinuous Galerkin methods for strongly nonlinear elliptic boundary value problems. Numer. Math. 109 (2008), 233-268. DOI 10.1007/s00211-008-0137-y | MR 2385653 | Zbl 1146.65076
[25] T. Gudi, K. Porwal: A posteriori error control of discontinuous Galerkin methods for elliptic obstacle problems. Math. Comput. 83 (2014), 579-602. DOI 10.1090/S0025-5718-2013-02728-7 | MR 3143685 | Zbl 1305.65231
[26] M. Hintermüller, K. Ito, K. Kunisch: The primal-dual active set strategy as a semismooth Newton method. SIAM J. Optim. 13 (2003), 865-888. DOI 10.1137/S1052623401383558 | MR 1972219 | Zbl 1080.90074
[27] J. Hozman, T. Tichý, M. Vlasák: DG method for pricing European options under Merton jump-diffusion model. Appl. Math., Praha 64 (2019), 501-530. DOI 10.21136/AM.2019.0305-18 | MR 4022161 | Zbl 07144726
[28] C. Johnson: A convergence estimate for an approximation of a parabolic variational inequality. SIAM J. Numer. Anal. 13 (1976), 599-606. DOI 10.1137/0713050 | MR 0483545 | Zbl 0337.65055
[29] D. Kinderlehrer, G. Stampacchia: An Introduction to Variational Inequalities and Their Applications. Classics in Applied Mathematics 31. SIAM, Philadelphia (2000). DOI 10.1137/1.9780898719451 | MR 1786735 | Zbl 0988.49003
[30] J.-L. Lions: Partial differential inequalities. Russ. Math. Surv. 27 (1972), 91-159. DOI 10.1070/RM1972v027n02ABEH001373 | MR 0296479 | Zbl 0246.35010
[31] J.-L. Lions, G. Stampacchia: Variational inequalities. Commun. Pure Appl. Math. 20 (1967), 493-519. DOI 10.1002/cpa.3160200302 | MR 0216344 | Zbl 0152.34601
[32] K.-S. Moon, R. H. Nochetto, T. von Petersdorff, C.-S. Zhang: A posteriori error analysis for parabolic variational inequalities. ESAIM, Math. Model. Numer. Anal. 41 (2007), 485-511. DOI 10.1051/m2an:2007029 | MR 2355709 | Zbl 1142.65053
[33] R. H. Nochetto, G. Savaré, C. Verdi: Error control of nonlinear evolution equations. C. R. Acad. Sci., Paris, Sér. I, Math. 326 (1998), 1437-1442. DOI 10.1016/S0764-4442(98)80407-2 | MR 1649189 | Zbl 0944.65077
[34] R. H. Nochetto, G. Savaré, C. Verdi: A posteriori error estimates for variable time-step discretizations of nonlinear evolution equations. Commun. Pure Appl. Math. 53 (2000), 525-589. DOI 10.1002/(SICI)1097-0312(200005)53:5<525::AID-CPA1>3.0.CO;2-M | MR 1737503 | Zbl 1021.65047
[35] E. Otárola, A. J. Salgado: Finite element approximation of the parabolic fractional obstacle problem. SIAM J. Numer. Anal. 54 (2016), 2619-2639. DOI 10.1137/15M1029801 | MR 3542012 | Zbl 1349.65473
[36] A. K. Pani, P. C. Das: A priori error estimates for a single-phase quasilinear Stefan problem in one space dimension. IMA J. Numer. Anal. 11 (1991), 377-392. DOI 10.1093/imanum/11.3.377 | MR 1118963 | Zbl 0727.65113
[37] B. Riviére: Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation. Frontiers in Applied Mathematics 35. SIAM, Philadelphia (2008). DOI 10.1137/1.9780898717440 | MR 2431403 | Zbl 1153.65112
[38] B. Riviére, M. F. Wheeler, V. Girault: A priori error estimates for finite element methods based on discontinuous approximation spaces for elliptic problems. SIAM J. Numer. Anal. 39 (2001), 902-931. DOI 10.1137/S003614290037174X | MR 1860450 | Zbl 1010.65045
[39] J. Rulla: Error analysis for implicit approximations to solutions to Cauchy problems. SIAM J. Numer. Anal. 33 (1996), 68-87. DOI 10.1137/0733005 | MR 1377244 | Zbl 0855.65102
[40] G. Savaré: Weak solutions and maximal regularity for abstract evolution inequalities. Adv. Math. Sci. Appl. 6 (1996), 377-418. MR 1411975 | Zbl 0858.35073
[41] V. Thomée: Galerkin Finite Element Methods for Parabolic Problems. Springer Series in Computational Mathematics 25. Springer, Berlin (2006). DOI 10.1007/3-540-33122-0 | MR 2249024 | Zbl 1105.65102
[42] C. Vuik: An $L^2$-error estimate for an approximation of the solution of a parabolic variational inequality. Numer. Math. 57 (1990), 453-471. DOI 10.1007/BF01386423 | MR 1063805 | Zbl 0696.65069
[43] M. F. Wheeler: An elliptic collocation-finite element method with interior penalties. SIAM J. Numer. Anal. 15 (1978), 152-161. DOI 10.1137/0715010 | MR 0471383 | Zbl 0384.65058
[44] X. Yang, G. Wang, X. Gu: Numerical solution for a parabolic obstacle problem with nonsmooth initial data. Numer. Methods Partial Differ. Equations 30 (2014), 1740-1754. DOI 10.1002/num.21893 | MR 3246191 | Zbl 1312.65107
[45] C.-S. Zhang: Adaptive Finite Element Methods for Variational Inequalities: Theory and Application in Finance: Ph.D. Thesis. University of Maryland, College Park (2007). MR 2711028

Affiliations:   Papri Majumder, Department of Mathematics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi - 110016, India, e-mail: pm9681@gmail.com


 
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