An L2-stability analysis of a θ-scheme for a class of nonlinear parabolic variational inequalities of obstacle type L2 Stability of a θ-scheme for parabolic PVIs
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Abstract
This paper analyzes the stability of a fully discrete finite element approximation for a class of nonlinear parabolic variational inequalities of obstacle type. The temporal discretization is based on a θ-scheme. We derive a stability condition for the scheme that depends critically on the parameter θ. We prove that the method is unconditionally stable in the L2-norm for θ in [1/2,1]. For θ in [0,1/2), we establish a precise Courant-Friedrichs-Lewy (CFL)-type condition, Delta t<=2gγ/(L2(1-2θ)), where γ is the coercivity constant and L is the Lipschitz constant of the nonlinear source term. The analysis is based on a careful choice of test functions in the variational inequality and by deriving sharp estimates of the associated bilinear form.
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References
[1] Y. Alnashri, A general error estimate for parabolic variational inequalities, Computational Methods in Applied Mathematics, 22(3) (2022), 245–258. DOI: 10.1515/cmam-2021-0050
[2] M. A. Bencheikh Le Hocine, S. Boulaaras, and M. Haiour, An optimal L∞L∞-error estimate for an approximation of a parabolic variational inequality, Numerical Functional Analysis and Optimization, 37(1) (2016), 1–18. DOI: 10.1080/01630563.2015.1109520
[3] A. Bensoussan and J. L. Lions, Problèmes de temps d'arrêt optimal et inéquations variationnelles paraboliques, Applicable Analysis, 3(3) (1973), 267–294. DOI: 10.1080/00036817308839070
[4] A. Bensoussan and J. L. Lions, Applications of Variational Inequalities in Stochastic Control, North-Holland, Amsterdam, 1982.
[5] S. Boulaaras and M. Haiour, The maximum norm analysis of an overlapping Schwarz method for parabolic quasi-variational inequalities related to impulse control problem with mixed boundary conditions, Applied Mathematics and Information Sciences, 7(1) (2013), 343–353. URL
[6] S. Boulaaras, M. A. Bencheikh Le Hocine, and M. Haiour, The finite element approximation in a system of parabolic quasi-variational inequalities related to management of energy production with mixed boundary condition, Computational Mathematics and Modeling, 25(4) (2014), 530–543. DOI: 10.1007/s10598-014-9247-9
[7] M. Boulbrachene, P. Cortey-Dumont, and J. C. Miellou, L∞-error estimate for a class of semilinear elliptic variational inequalities and quasi-variational inequalities, International Journal of Mathematics and Mathematical Sciences, 27(5) (2001), 309–319. DOI: 10.1155/S0161171201010602
[8] M. Boulbrachene, Optimal L∞L∞-error estimate for variational inequalities with nonlinear source terms, Applied Mathematics Letters, 15(8) (2002), 1013–1017. DOI: 10.1016/S0893-9659(02)00078-2
[9] M. Boulbrachene, Pointwise error estimates for a class of elliptic quasi-variational inequalities with nonlinear source terms, Applied Mathematics and Computation, 161(1) (2005), 129–138. DOI: 10.1016/j.amc.2003.12.015
[10] M. Boulbrachene, On the finite element approximation of variational inequalities with noncoercive operators, Numerical Functional Analysis and Optimization, 36(9) (2015), 1107–1121. DOI: 10.1080/01630563.2015.1056913
[11] M. Boulbrachene, M. Haiour, and B. Chentouf, On a noncoercive system of quasi-variational inequalities related to stochastic control problems, Journal of Inequalities in Pure and Applied Mathematics, 3(2) (2002), Article 25. URL
[12] H. Brézis and G. Stampacchia, Sur la régularité de la solution d'inéquation elliptique, Bulletin de la Société Mathématique de France, 96 (1968), 153–180. DOI: 10.24033/bsmf.1663
[13] J. J. Carvajal, D. Damircheli, T. Führer, F. Fuica, and M. Karkulik, A space-time finite element method for parabolic obstacle problems, (2025), preprint. DOI: 10.48550/arXiv.2503.07808
[14] P. Cortey-Dumont, On finite element approximation in the L∞L∞-norm of variational inequalities, Numerische Mathematik, 47 (1985), 45–57. DOI: 10.1007/BF01389875
[15] G. Duvaut and J.-L. Lions, Les inéquations en mécanique et en physique, Dunod, Paris, 1972.
[16] H. Falk, Error estimates for the approximation of a class of variational inequalities, Mathematics of Computation, 28(128) (1974), 963–971. DOI: 10.2307/2005358
[17] A. Friedman, Parabolic variational inequalities in one space dimension and smoothness of the free boundary, Journal of Functional Analysis, 18(2) (1975), 151–176. DOI: 10.1016/0022-1236(75)90022-1
[18] R. Glowinski, Numerical Methods for Nonlinear Variational Problems, Springer-Verlag, New York, 1984. URL
[19] T. Gustafsson, Adaptive finite elements for obstacle problems, Advances in Applied Mechanics, 58 (2024), 205–243. DOI: 10.1016/bs.aams.2024.03.004
[20] M. Haiour and E. Hadidi, Uniform convergence of Schwarz method for noncoercive variational inequalities, International Journal of Contemporary Mathematical Sciences, 4(29) (2009), 1423–1434.
[21] W. Han and C. Wang, Numerical analysis of a parabolic hemivariational inequality for semipermeable media, Journal of Computational and Applied Mathematics, 389 (2021), 113326. DOI: 10.1016/j.cam.2020.113326
[22] J. Kacur and R. Van Keer, Solution of degenerate parabolic variational inequalities with convection, ESAIM: Mathematical Modelling and Numerical Analysis, 37(3) (2003), 417–431. DOI: 10.1051/m2an:2003035
[23] J. Li and C. Bi, Study of weak solutions of variational inequality systems with quasilinear degenerate parabolic operators, AIMS Mathematics, 7(11) (2022), 19758–19769. DOI: 10.3934/math.20221083
[24] J. L. Lions and G. Stampacchia, Variational inequalities, Communications on Pure and Applied Mathematics, 20(3) (1967), 493–519. DOI: 10.1002/cpa.3160200302
[25] S. Madi, M. C. Bouras, M. Haiour, and A. Stahel, Pricing of American options using the Brennan–Schwartz algorithm based on finite elements, Applied Mathematics and Computation, 339 (2018), 846–852. DOI: 10.24451/arbor.6773
[26] R. H. Nochetto, K. G. Siebert, and A. Veeser, Theory of adaptive finite element methods: An introduction, in: Multiscale, Nonlinear and Adaptive Approximation, Springer, Berlin, 2009, pp. 409–542. DOI: 10.1007/978-3-642-03413-8_12
[27] J. F. Rodrigues, Obstacle Problems in Mathematical Physics, North-Holland, Amsterdam, 1987.
[28] G. Stampacchia, Formes bilinéaires coercitives sur les ensembles convexes, Comptes Rendus de l'Académie des Sciences, 258 (1964), 4413–4416.
[29] C. Vuik, An L2-error estimate for an approximation of the solution of a parabolic variational inequality, Numerische Mathematik, 57 (1990), 453–471. DOI: 10.1007/BF01386423