A quasi-optimal error estimate for a discrete singularly perturbed approximation to the prescribed curvature problem
HTML articles powered by AMS MathViewer
- by Maurizio Paolini PDF
- Math. Comp. 66 (1997), 45-67 Request permission
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.References
- 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
- G. Barles, H. M. Soner, and P. E. Souganidis, Front propagation and phase field theory, SIAM J. Control Optim. 31 (1993), no. 2, 439–469. MR 1205984, DOI 10.1137/0331021
- G. Bellettini and M. Paolini, Quasi-optimal error estimates for the mean curvature flow with a forcing term, Differential Integral Equations 8 (1995), no. 4, 735–752. MR 1306590
- Giovanni Bellettini, Maurizio Paolini, and Claudio 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), no. 4, 317–328 (English, with Italian summary). MR 1096825
- G. Bellettini, M. Paolini, and C. Verdi, Numerical minimization of geometrical type problems related to calculus of variations, Calcolo 27 (1990), no. 3-4, 251–278 (1991). MR 1141029, DOI 10.1007/BF02575797
- E. Bombieri, E. De Giorgi, and E. Giusti, Minimal cones and the Bernstein problem, Invent. Math. 7 (1969), 243–268. MR 250205, DOI 10.1007/BF01404309
- Lia Bronsard and Robert V. Kohn, Motion by mean curvature as the singular limit of Ginzburg-Landau dynamics, J. Differential Equations 90 (1991), no. 2, 211–237. MR 1101239, DOI 10.1016/0022-0396(91)90147-2
- Xinfu Chen, Generation and propagation of interfaces for reaction-diffusion equations, J. Differential Equations 96 (1992), no. 1, 116–141. MR 1153311, DOI 10.1016/0022-0396(92)90146-E
- Xinfu Chen and Charles M. Elliott, Asymptotics for a parabolic double obstacle problem, Proc. Roy. Soc. London Ser. A 444 (1994), no. 1922, 429–445. MR 1290089, DOI 10.1098/rspa.1994.0030
- Philippe G. Ciarlet, The finite element method for elliptic problems, Studies in Mathematics and its Applications, Vol. 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. MR 0520174
- Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
- 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.
- Ennio De Giorgi and Tullio Franzoni, Su un tipo di convergenza variazionale, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8) 58 (1975), no. 6, 842–850 (Italian). MR 448194
- P. De Mottoni and M. Schatzman, Geometrical evolution of developed interfaces, Trans. Amer. Math. Soc. 347 (1995), 1533–1589
- G. Dziuk and J.E. Hutchinson, On the approximation of unstable parametric minimal surfaces, Calc. Var. Partial Differential Equations 4 (1996), 27–58.
- L. C. Evans, H. M. Soner, and P. E. Souganidis, Phase transitions and generalized motion by mean curvature, Comm. Pure Appl. Math. 45 (1992), no. 9, 1097–1123. MR 1177477, DOI 10.1002/cpa.3160450903
- Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York, Inc., New York, 1969. MR 0257325
- Robert Finn, Equilibrium capillary surfaces, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 284, Springer-Verlag, New York, 1986. MR 816345, DOI 10.1007/978-1-4613-8584-4
- Avner Friedman, Variational principles and free-boundary problems, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1982. MR 679313
- David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190, DOI 10.1007/978-3-642-61798-0
- Enrico Giusti, Minimal surfaces and functions of bounded variation, Monographs in Mathematics, vol. 80, Birkhäuser Verlag, Basel, 1984. MR 775682, DOI 10.1007/978-1-4684-9486-0
- Sigurdur Helgason, Groups and geometric analysis, Pure and Applied Mathematics, vol. 113, Academic Press, Inc., Orlando, FL, 1984. Integral geometry, invariant differential operators, and spherical functions. MR 754767
- Tom Ilmanen, Convergence of the Allen-Cahn equation to Brakke’s motion by mean curvature, J. Differential Geom. 38 (1993), no. 2, 417–461. MR 1237490
- H. Lewy, Aspects of the Calculus of Variations, University of California Press, Berkeley, 1939
- Luciano Modica and Stefano Mortola, Un esempio di $\Gamma ^{-}$-convergenza, Boll. Un. Mat. Ital. B (5) 14 (1977), no. 1, 285–299 (Italian, with English summary). MR 0445362
- Joachim Nitsche, $L_{\infty }$-convergence of finite element approximations, Mathematical aspects of finite element methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975) Lecture Notes in Math., Vol. 606, Springer, Berlin, 1977, pp. 261–274. MR 0488848
- Johannes C. C. Nitsche, Lectures on minimal surfaces. Vol. 1, Cambridge University Press, Cambridge, 1989. Introduction, fundamentals, geometry and basic boundary value problems; Translated from the German by Jerry M. Feinberg; With a German foreword. MR 1015936
- Ricardo H. Nochetto, Maurizio Paolini, and Claudio Verdi, Sharp error analysis for curvature dependent evolving fronts, Math. Models Methods Appl. Sci. 3 (1993), no. 6, 711–723. MR 1245632, DOI 10.1142/S0218202593000369
- R. H. Nochetto, M. Paolini, and C. Verdi, Optimal interface error estimates for the mean curvature flow, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 21 (1994), no. 2, 193–212. MR 1288364
- R. H. Nochetto, M. Paolini, and C. Verdi, Quadratic rate of convergence for curvature dependent smooth interfaces: a simple proof, Appl. Math. Lett. 7 (1994), no. 4, 59–63. MR 1350394, DOI 10.1016/0893-9659(94)90012-4
- R. H. Nochetto, M. Paolini, and C. Verdi, Double obstacle formulation with variable relaxation parameter for smooth geometric front evolutions: asymptotic interface error estimates, Asymptotic Anal. 10 (1995), no. 2, 173–198. MR 1324387
- —, A dynamic mesh algorithm for curvature dependent evolving interfaces, J. Comput. Phys. 123 (1996), 296–310.
- M. Paolini and C. Verdi, Asymptotic and numerical analyses of the mean curvature flow with a space-dependent relaxation parameter, Asymptotic Anal. 5 (1992), no. 6, 553–574. MR 1169358
- Rolf Rannacher and Ridgway Scott, Some optimal error estimates for piecewise linear finite element approximations, Math. Comp. 38 (1982), no. 158, 437–445. MR 645661, DOI 10.1090/S0025-5718-1982-0645661-4
- Ridgway Scott, Optimal $L^{\infty }$ estimates for the finite element method on irregular meshes, Math. Comp. 30 (1976), no. 136, 681–697. MR 436617, DOI 10.1090/S0025-5718-1976-0436617-2
- James Simons, Minimal varieties in riemannian manifolds, Ann. of Math. (2) 88 (1968), 62–105. MR 233295, DOI 10.2307/1970556
- Joel Smoller, Shock waves and reaction-diffusion equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 258, Springer-Verlag, New York-Berlin, 1983. MR 688146
Additional Information
- Maurizio Paolini
- Affiliation: Dipartimento di Matematica e Informatica, Università di Udine, 33100, Udine, Italy
- Email: paolini@dimi.uniud.it
- 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.
- © Copyright 1997 American Mathematical Society
- 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