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A quasi-optimal error estimate for a discrete
singularly perturbed approximation to the
prescribed curvature problem


Author: Maurizio Paolini
Journal: Math. Comp. 66 (1997), 45-67
MSC (1991): Primary 35B25, 35J60, 65N30; Secondary 35A35, 49Q05
DOI: https://doi.org/10.1090/S0025-5718-97-00771-0
MathSciNet review: 1361810
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Abstract: Solutions of the so-called prescribed curvature problem
$\min _{A\subseteq \Omega } \mathcal {P}_ \Omega (A) - \int _A g(x)$, $g$ being the curvature field, are approximated via a singularly perturbed elliptic PDE of bistable type. For nondegenerate relative minimizers $A \subset \subset \Omega $ we prove an $ \mathcal {O}( \epsilon ^2 |\log \epsilon |^2)$ error estimate (where $ \epsilon $ stands for the perturbation parameter), and show that this estimate is quasi-optimal. The proof is based on the construction of accurate barriers suggested by formal asymptotics. This analysis is next extended to a finite element discretization of the PDE to prove the same error estimate for discrete minima.


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  • [1] S.M. Allen and J.W. Cahn, A macroscopic theory for antiphase boundary motion and its application to antiphase domain coarsing, Acta Metall. 27 (1979), 1085-1095
  • [2] G. Barles, H.-M. Soner, and P.E. Souganidis, Front propagation and phase field theory, SIAM J. Control Optim. 31 (1993), 439-469. MR 94c:35005
  • [3] G. Bellettini and M. Paolini, Quasi-optimal error estimates for the mean curvature flow with a forcing term, Differential Integral Equations 8 (1995), 735-752. MR 95h:35021
  • [4] G. Bellettini, M. Paolini, and C. Verdi, $\Gamma $-convergence of discrete approximations to interfaces with prescribed mean curvature, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 1 (1990), 317-328. MR 92a:49019
  • [5] -, Numerical minimization of geometrical type problems related to calculus of variations, Calcolo 27 (1990), 251-278. MR 92i:65103
  • [6] E. Bombieri, E. De Giorgi, and E. Giusti, Minimal cones and the Bernstein problem, Invent. Math. 7 (1969), 243-268. MR 40:3445
  • [7] L. Bronsard and R.V. Kohn, Motion by mean curvature as the singular limit of Ginzburg-Landau dynamics, J. Differential Equations 90 (1991), 211-237. MR 92d:35037
  • [8] X. Chen, Generation and propagation of interfaces in reaction-diffusion equations, J. Differential Equations 96 (1992), 116-141. MR 92m:35129
  • [9] X. Chen and C.M. Elliott, Asymptotics for a parabolic double obstacle problem, Proc. Roy. Soc. London Ser. A 444 (1994), 429-445. MR 95f:35128
  • [10] P.G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978. MR 58:25001
  • [11] R. Courant and D. Hilbert, Methods of Mathematical Physics, Volume I, Interscience Publishers Inc., New York, 1953. MR 16:426a
  • [12] E. De Giorgi, Congetture sui limiti delle soluzioni di alcune equazioni paraboliche quasi lineari, Nonlinear Analysis. A Tribute in Honour of G. Prodi, S.N.S. Quaderni, Pisa, 1991, pp. 173-187.
  • [13] E. De Giorgi and T. Franzoni, Su un tipo di convergenza variazionale, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 58 (1975), 842-850. MR 56:6503
  • [14] P. De Mottoni and M. Schatzman, Geometrical evolution of developed interfaces, Trans. Amer. Math. Soc. 347 (1995), 1533-1589
  • [15] G. Dziuk and J.E. Hutchinson, On the approximation of unstable parametric minimal surfaces, Calc. Var. Partial Differential Equations 4 (1996), 27-58. CMP 96:09
  • [16] L.C. Evans, H.-M. Soner, and P.E. Souganidis, Phase transitions and generalized motion by mean curvature, Comm. Pure Appl. Math. 45 (1992), 1097-1123. MR 93g:35064
  • [17] H. Federer, Geometric Measure Theory, Springer- Verlag, Berlin, 1968. MR 41:1976
  • [18] R. Finn, Equilibrium Capillary Surfaces, Springer-Verlag, Berlin, 1986. MR 88f:49001
  • [19] A. Friedman, Variational Principles and Free Boundary Problems, Wiley, New York, 1982. MR 84e:35153
  • [20] D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1983. MR 86c:35035
  • [21] E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Birkhäuser, Boston, 1984. MR 87a:58041
  • [22] S. Helgason, Groups and Geometric Analysis, Academic Press, Orlando, Florida, 1984. MR 86c:22017
  • [23] T. Ilmanen, Convergence of the Allen-Cahn equation to Brakke's motion by mean curvature, J. Differential Geom. 38 (1993), 417-461. MR 94h:58051
  • [24] H. Lewy, Aspects of the Calculus of Variations, University of California Press, Berkeley, 1939
  • [25] L. Modica and S. Mortola, Un esempio di $\Gamma $-convergenza, Boll. Un. Mat. Ital. B (5) 14 (1977), 285-299. MR 56:3704
  • [26] J. Nitsche, $L_\infty $-convergence of finite element approximations, Mathematical Aspects of Finite Element Methods, Lecture Notes in Math., 606., Springer-Verlag, Berlin, 1977, pp. 261-274. MR 58:8351
  • [27] J.C.C. Nitsche, Lectures on Minimal Surfaces, Volume 1, Cambridge University Press, Cambridge, 1989. MR 90m:49031
  • [28] R.H. Nochetto, M. Paolini, and C. Verdi, Sharp error analysis for curvature dependent evolving fronts, Math. Models Methods Appl. Sci. 3 (1993), 711-723. MR 94k:35028
  • [29] -, Optimal interface error estimates for the mean curvature flow, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 21 (1994), 193-212. MR 96b:35020
  • [30] -, Quadratic rate of convergence for curvature dependent smooth interfaces: a simple proof, Appl. Math. Lett. 7 (1994), 59-63. MR 96d:35067
  • [31] -, Double obstacle formulation with variable relaxation parameter for smooth geometric front evolutions: asymptotic interface error estimates, Asymptotic Anal. 10 (1995), 173-198. MR 96b:35238
  • [32] -, A dynamic mesh algorithm for curvature dependent evolving interfaces, J. Comput. Phys. 123 (1996), 296-310. CMP 96:07
  • [33] M. Paolini and C. Verdi, Asymptotic and numerical analyses of the mean curvature flow with a space-dependent relaxation parameter, Asymptotic Anal. 5 (1992), 553-574. MR 93h:35101
  • [34] R. Rannacher and R. Scott, Some optimal error estimates for piecewise linear finite element approximations, Math. Comp. 38 (1982), 437-445. MR 83e:65180
  • [35] R. Scott, Optimal $L^\infty $ estimates for the finite element method on irregular meshes, Math. Comp. 32 (1978), 681-697. MR 55:9560
  • [36] J. Simons, Minimal varieties in Riemannian manifolds, Ann. of Math. (2) 88 (1968), 62-105. MR 38:1617
  • [37] J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York, 1983. MR 84d:35002

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Additional Information

Maurizio Paolini
Affiliation: Dipartimento di Matematica e Informatica, Università di Udine, 33100, Udine, Italy
Email: paolini@dimi.uniud.it

DOI: https://doi.org/10.1090/S0025-5718-97-00771-0
Received by editor(s): September 28, 1994
Received by editor(s) in revised form: August 9, 1995
Additional Notes: This work was partially supported by MURST (Progetto Nazionale “Equazioni di Evoluzione e Applicazioni Fisico-Matematiche” and “Analisi Numerica e Matematica Computazionale”) and CNR (IAN and Contracts 92.00833.01, 93.00564.01) of Italy.
Article copyright: © Copyright 1997 American Mathematical Society

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