bdim: Biblioteca Digitale Italiana di Matematica

Un progetto SIMAI e UMI

Referenza completa

Verdi, Claudio:
Numerical methods for phase transition problems
Bollettino dell'Unione Matematica Italiana Serie 8 1-B (1998), fasc. n.1, p. 83-108, Unione Matematica Italiana (English)
pdf (363 Kb), djvu (404 Kb). | MR1619039 | Zbl 0896.65064

Sunto

Nel presente articolo si illustrano alcuni dei principali metodi numerici per l'approssimazione di modelli matematici legati ai fenomeni di transizione di fase. Per semplificare e contenere l'esposizione ci siamo limitati a discutere con un certo dettaglio i metodi più recenti, presentandoli nel caso di problemi modello, quali il classico problema di Stefan e l'evoluzione di superficie per curvatura media, solo accennando alle applicazioni e modelli più generali.
Referenze Bibliografiche
[1] S. M. ALLEN - J. W. CAHN , A macroscopic theory for antiphase boundary motion and its application to antiphase domain coarsing, Acta Metall. Mater., 27 (1979), 1085-1095.
[2] F. ALMGREN - J. E. TAYLOR - L. WANG , Curvature-driven flows: a variational approach, SIAM J. Control Optim., 31 (1993), 387-438. | MR 1205983 | Zbl 0783.35002
[3] R. ALMGREN , Variational algorithms and pattern formation in dendritic solidification, J. Comput. Phys., 106 (1993), 337-354. | MR 1218734 | Zbl 0787.65095
[4] S. ALTSCHULER - S. B. ANGENENT - Y. GIGA , Mean curvature flow through singularities for surfaces of rotation, J. Geom. Anal., 5 (1995), 293-358. | MR 1360824 | Zbl 0847.58072
[5] I. ATHANASOPOULOS - L. CAFFARELLI - S. SALSA , Degenerate phase transition problems of parabolic type. Smoothness of the front, to appear. | Zbl 0924.35197
[6] G. BARLES - H. M. SONER - P. E. SOUGANIDIS , Front propagation and phase field theory, SIAM J. Control Optim., 31 (1993), 439-469. | MR 1205984 | Zbl 0785.35049
[7] G. BELLETTINI - M. PAOLINI , Two examples of fattening for the curvature flow with a driving force, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. (9) Mat. Appl., 5 (1994), 229-236. | MR 1298266 | Zbl 0826.35051
[8] G. BELLETTINI - M. PAOLINI , Quasi-optimal error estimates for the mean curvature flow with a forcing term, Diff. Integ. Eq., 8 (1995), 735-752. | MR 1306590 | Zbl 0820.49019
[9] G. BELLETTINI - M. PAOLINI , Some results on minimal barriers in the sense of De Giorgi applied to driven motion by mean curvature, Rend. Accad. Naz. Sci. XL, Mem. Mat. (5), 19 (1995), 43-67. | MR 1387549 | Zbl 0944.53039
[10] G. BELLETTINI - M. PAOLINI , Anisotropic motion by mean curvature in the context of Finsler geometry, Hokkaido Math. J., 25 (1996), 537-566. | MR 1416006 | Zbl 0873.53011
[11] G. BELLETTINI - M. PAOLINI - C. VERDI , Front-tracking and variational methods to approximate interfaces with prescribed mean curvature, in Numerical Methods for Free Boundary Problems (P. NEITTAANMÄKI ed.), Birkhäuser, Basel (1991), pp. 83-92. | MR 1118855 | Zbl 0754.65065
[12] G. BELLETTINI - M. PAOLINI - C. VERDI , Convex approximations of functionals with curvature, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. (9) Mat. Appl., 2 (1991), 297-306. | MR 1152636 | Zbl 0754.65066
[13] J. F. BLOWEY - C. M. ELLIOTT , A phase-field model with a double obstacle potential, in Motion by Mean Curvature and Related Topics (G. BUTTAZZO and A. VISINTIN eds.), Gruyter, New York, (1994), pp. 1-22. | MR 1277388 | Zbl 0809.35168
[14] F. BREZZI - L. A. CAFFARELLI , Convergence of the discrete free boundaries for finite element approximations, RAIRO Modél. Math. Anal. Numér., 17 (1983), 385-395. | fulltext mini-dml | MR 713766 | Zbl 0547.65081
[15] E. BÄNSCH , Local mesh refinement in 2 and 3 dimensions, IMPACT Comput. Sci. Engrg., 3 (1991), 181-191. | MR 1141298 | Zbl 0744.65074
[16] G. CAGINALP , An analysis of a phase field model of a free boundary, Arch. Rational Mech. Anal., 92 (1986), 205-245. | MR 816623 | Zbl 0608.35080
[17] X. CHEN - F. REITICH , Local existence and uniqueness of solutions of the Stefan problem with surface tension and kinetic undercooling, J. Math. Anal. Appl., 164 (1992), 350-362. | MR 1151039 | Zbl 0761.35113
[18] Y. G. CHEN - Y. GIGA - S. GOTO , Uniqueness and existence of viscosity solutions of generalized mean curvature flow equation, J. Diff. Geom., 33 (1991), 749-786. | fulltext mini-dml | MR 1100211 | Zbl 0696.35087
[19] Z. CHEN - R. H. NOCHETTO , A posteriori error estimation for the continuous casting problem, in preparation.
[20] Z. CHEN - R. H. NOCHETTO , A posteriori error estimation and adaptivity for phase relaxation models, in preparation.
[21] P. G. CIARLET , The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam (1978). | MR 520174 | Zbl 0383.65058
[22] H. CLÉMENT , Approximation by finite element functions using local regularization, RAIRO Modél. Math. Anal. Numér., 9 (1975), 77-84. | fulltext mini-dml | MR 400739 | Zbl 0368.65008
[23] M. G. CRANDALL - H. ISHII - P. L. LIONS , User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1-67. | fulltext mini-dml | MR 1118699 | Zbl 0755.35015
[24] G. DAL MASO , An Introduction to G-Convergence, Birkhäuser, Boston (1993). | MR 1201152 | Zbl 0816.49001
[25] T. DUPONT , Mesh modification for evolution equations, Math. Comp., 29 (1982), 85-107. | MR 658215 | Zbl 0493.65044
[26] G. DZIUK , An algorithm for evolutionary surfaces, Numer. Math., 58 (1991), 603-611. | MR 1083523 | Zbl 0714.65092
[27] G. DZIUK , Convergence of a semi-discrete scheme for the curvature shortening flow, Math. Models Methods Appl. Sci., 4 (1994), 589-606. | MR 1291140 | Zbl 0811.65112
[28] G. DZIUK , Convergence of a semi-discrete scheme for the anisotropic curvature shortening flow, to appear.
[29] C. M. ELLIOTT , Error analysis of the enthalpy method for the Stefan problem, IMA J. Numer. Anal., 7 (1987), 61-71. | MR 967835 | Zbl 0638.65088
[30] C. M. ELLIOTT - M. PAOLINI - R. SCHÄTZLE , Interface estimates for the fully anisotropic Allen-Cahn equation and anisotropic mean curvature flow, Math. Models Methods Appl. Sci., 6 (1996), 1103-1118. | MR 1428147 | Zbl 0873.35039
[31] C. M. ELLIOTT - R. SCHÄTZLE , The limit of the anisotropic double-obstacle Allen-Cahn equation, Proc. Roy. Soc. London Ser. A, to appear. | MR 1424223 | Zbl 0865.35073
[32] C. M. ELLIOTT - R. SCHÄTZLE , The limit of the fully anisotropic double-obstacle Allen-Cahn equation in the non-smooth case, SIAM J. Math. Anal., to appear. | MR 1434036 | Zbl 0870.35128
[33] K. ERIKSSON - C. JOHNSON , Adaptive finite element methods for parabolic problems I: a linear model problem, SIAM J. Numer. Anal., 28 (1991), 43-77. | MR 1083324 | Zbl 0732.65093
[34] L. C. EVANS - H. M. SONER - P. E. SOUGANIDIS , Phase transitions and generalized motion by mean curvature, Comm. Pure Appl. Math., 45 (1992), 1097-1123. | MR 1177477 | Zbl 0801.35045
[35] L. C. EVANS - J. SPRUCK , Motion of level sets by mean curvature. I, J. Diff. Geom., 33 (1991), 635-681. | fulltext mini-dml | MR 1100206 | Zbl 0726.53029
[36] F. FIERRO , Numerical approximation for the mean curvature flow with nucleation using implicit time-stepping: an adaptive algorithm, Calcolo, to appear. | MR 1740750 | Zbl 0927.65144
[37] F. FIERRO - R. GOGLIONE - M. PAOLINI , Numerical simulations of mean curvature flow in presence of a nonconvex anisotropy, Math. Models Methods Appl. Sci., to appear. | MR 1634826 | Zbl 0946.58014
[38] P. C. FIFE - O. PENROSE , Thermodynamically consistent models of phase-field type for the kinetics of phase transitions, Physica D, 43 (1990), 44-62. | MR 1060043 | Zbl 0709.76001
[39] Y. GIGA - S. GOTO - H. ISHII - M. H. SATO , Comparison principle and convexity preserving properties for singular degenerate parabolic equations on unbounded domains, Indiana Univ. Math. J., 40 (1991), 443-470. | MR 1119185 | Zbl 0836.35009
[40] G. HUISKEN , Flow by mean curvature of convex surfaces into spheres, J. Diff. Geom., 20 (1994), 237-266. | fulltext mini-dml | MR 772132 | Zbl 0556.53001
[41] J. W. JEROME - M. E. ROSE , Error estimates for the multidimensional two-phase Stefan problem, Math. Comp., 39 (1982), 377-414. | MR 669635 | Zbl 0505.65060
[42] X. JIANG - R. H. NOCHETTO , A finite element method for a phase relaxation model. Part I: quasi-uniform mesh, SIAM J. Numer. Anal., to appear. | MR 1619875 | Zbl 0972.65067
[43] X. JIANG - R. H. NOCHETTO - C. VERDI , A P 12P0 finite element method for a model of polymer crystallization, Comput. Meth. Appl. Mech. Engrg., 125 (1995), 303-317. | MR 1352100 | Zbl 0949.82025
[44] X. JIANG - R. H. NOCHETTO - C. VERDI , A P 12P1 finite element method for a phase relaxation model. Part II: adaptively refined meshes, SIAM J. Numer. Anal., to appear. | MR 1688994 | Zbl 0934.65105
[45] M. KIMURA , Numerical analysis for moving boundary problems using the boundary tracking method, Japan J. Indust. Appl. Math., to appear. | MR 1475140 | Zbl 0892.76065
[46] O. W. KLEIN , Existence and approximation results for phase-field systems of Penrose-Fife type and Stefan problems, Ph.D. Thesis, Humboldt-Universität, Berlin (1997).
[47] R. KORNHUBER , Adaptive Monotone Multigrid Methods for Nonlinear Variational Problems, Teubner Stuttgart (1997). | MR 1469497 | Zbl 0879.65041
[48] O. A. LADYZENSKAJA - V. SOLONNIKOV - N. URAL'CEVA , Linear and Quasilinear Equations of Parabolic Type, vol. TMM 23, AMS, Providence (1968). | MR 241822 | Zbl 0174.15403
[49] S. LUCKHAUS , Solutions for the two-phase Stefan problem with the Gibbs-Thomson law for the melting temperature, European J. Appl. Math., 1 (1990), 101-111. | MR 1117346 | Zbl 0734.35159
[50] S. LUCKHAUS - T. STURZENHECKER , Implicit time discretization for the mean curvature flow equation, Calc. Var. Partial Differential Equations, 3 (1995), 253-271. | MR 1386964 | Zbl 0821.35003
[51] E. MAGENES - R. H. NOCHETTO - C. VERDI , Energy error estimates for a linear scheme to approximate nonlinear parabolic problems, RAIRO Modél. Math. Anal. Numér., 21 (1987), 655-678. | fulltext mini-dml | MR 921832 | Zbl 0635.65123
[52] R. H. NOCHETTO , Error estimates for multidimensional singular parabolic problems, Japan J. Indust. Appl. Math., 4 (1987), 111-138. | MR 899207 | Zbl 0657.65132
[53] R. H. NOCHETTO , A stable extrapolation method for multidimensional degenerate parabolic problems, Math. Comp., 53 (1989), 455-470. | MR 982372 | Zbl 0675.65112
[54] R. H. NOCHETTO , Finite element methods for parabolic free boundary problems, in Advances in Numerical Analysis, vol. I: Nonlinear Partial Differential Equations and Dynamical Systems (W. LIGHT ed.), Oxford University Press, Oxford (1991), pp. 34-88. | MR 1138471 | Zbl 0733.65089
[55] R. H. NOCHETTO - M. PAOLINI - C. VERDI , An adaptive finite elements method for two-phase Stefan problems in two space dimensions. Part I: stability and error estimates. Supplement, Math. Comp., 57 (1991), 73-108, S1-S11. | MR 1079028 | Zbl 0733.65087
[56] R. H. NOCHETTO - M. PAOLINI - C. VERDI , An adaptive finite elements method for twophase Stefan problems in two space dimensions. Part II: implementation and numerical experiments, SIAM J. Sci. Statist. Comput., 12 (1991), 1207-1244. | MR 1114983 | Zbl 0733.65088
[57] R. H. NOCHETTO - M. PAOLINI - C. VERDI , A fully discrete adaptive nonlinear Chernoff formula, SIAM J. Numer. Anal., 30 (1993), 991-1014. | MR 1231324 | Zbl 0805.65135
[58] R. H. NOCHETTO - M. PAOLINI - C. VERDI , Sharp error analysis for curvature dependent evolving fronts, Math. Models Methods Appl. Sci., 3 (1993), 711-723. | MR 1245632 | Zbl 0802.65124
[59] R. H. NOCHETTO - M. PAOLINI - C. VERDI , Optimal interface error estimates for the mean curvature flow, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 21 (1994), 193-212. | fulltext mini-dml | MR 1288364 | Zbl 0886.35079
[60] R. H. NOCHETTO - M. PAOLINI - C. VERDI , Double obstacle formulation with variable relaxation parameter for smooth geometric front evolutions: asymptotic interface error estimates, Asymptotic Anal., 10 (1995), 173-198. | MR 1324387 | Zbl 0852.35060
[61] R. H. NOCHETTO - M. PAOLINI - C. VERDI , A dynamic mesh method for curvature dependent evolving interfaces, J. Comput. Phys., 123 (1996), 296-310. | MR 1372375 | Zbl 0851.65067
[62] R. H. NOCHETTO - M. PAOLINI - C. VERDI , Numerical Analysis of Geometric Motion of Fronts, CRM, Montreal, in preparation.
[63] R. H. NOCHETTO - A. SCHMIDT - C. VERDI , A posteriori error estimation and adaptivity for degenerate parabolic problems, Math. Comp., to appear. | MR 1648399 | Zbl 0942.65111
[64] R. H. NOCHETTO - A. SCHMIDT - C. VERDI , Mesh and time step modification for degenerate parabolic problems, in preparation.
[65] R. H. NOCHETTO - A. SCHMIDT - C. VERDI , Adaptive algorithm and simulations for Stefan problems in two and three dimensions, in preparation.
[66] R. H. NOCHETTO - A. SCHMIDT - C. VERDI , Adapting meshes and time-steps for phase change problems, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. (9) Mat. Appl., to appear. | MR 1631617 | Zbl 0910.65106
[67] R. H. NOCHETTO - C. VERDI , Approximation of degenerate parabolic problems using numerical integration, SIAM J. Numer. Anal., 25 (1988), 784-814. | MR 954786 | Zbl 0655.65131
[68] R. H. NOCHETTO - C. VERDI , An efficient linear scheme to approximate parabolic free boundary problems: error estimates and implementation, Math. Comp., 51 (1988), 27-53. | MR 942142 | Zbl 0657.65131
[69] R. H. NOCHETTO - C. VERDI , Convergence of double obstacle problems to the generalized geometric motion of fronts, SIAM J. Math. Anal., 26 (1995), 1514-1526. | MR 1356457 | Zbl 0839.35008
[70] R. H. NOCHETTO - C. VERDI , Approximating curvature driven interfaces with applications to shape recovery, in Curvature Flows and Related Topics (A. DAMLAMIAN et al., eds.), Gakkötosho, Tokyo (1995), pp. 159-177. | MR 1365307 | Zbl 0844.76007
[71] R. H. NOCHETTO - C. VERDI , Combined effect of explicit time-stepping and quadrature for curvature driven flows, Numer. Math., 74 (1996), 105-136 | MR 1400218 | Zbl 0859.65066
[72] R. H. NOCHETTO - C. VERDI , Convergence past singularities for a fully discrete approximation of curvature driven interfaces, SIAM J. Numer. Anal., 34 (1997), 490-512. | MR 1442924 | Zbl 0876.35053
[73] S. OSHER - J. A. SETHIAN , Fronts propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., 79 (1988), 12-49. | MR 965860 | Zbl 0659.65132
[74] M. PAOLINI , An efficient algorithm for computing anisotropic evolution by mean curvature, in Curvature Flows and Related Topics (A. DAMLAMIAN et al. eds.), Gakkötosho, Tokyo (1995), pp. 119-213. | MR 1365309 | Zbl 0838.73079
[75] M. PAOLINI - C. VERDI , Asymptotic and numerical analyses of the mean curvature flow with a space-dependent relaxation parameter, Asymptotic Anal., 5 (1992), 553-574. | MR 1169358 | Zbl 0757.65078
[76] J. RULLA - N. J. WALKINGTON , Optimal rates of convergence for degenerate parabolic problems in two dimensions, SIAM J. Numer. Anal., 33 (1996), 56-67. | MR 1377243 | Zbl 0856.65102
[77] A. SCHMIDT , Computation of three dimensional dendrites with finite elements, J. Comput. Phys., 125 (1996), 293-312. | Zbl 0844.65096
[78] J. A. SETHIAN , Level Set Methods, Cambridge University Press, Cambridge (1996). | MR 1409367 | Zbl 0859.76004
[79] C. VERDI , Optimal error estimates for an approximation of degenerate parabolic problems, Numer. Funct. Anal. Optim., 9 (1987), 657-670. | MR 895990 | Zbl 0598.65091
[80] C. VERDI , Numerical aspects of parabolic free boundary and hysteresis problems, in Phase Transition and Hysteresis (A. VISINTIN ed.), Lectures Notes in Mathematics, 1584, Springer-Verlag, Berlin (1994), pp. 213-284. | MR 1321834 | Zbl 0819.35155
[81] C. VERDI - A. VISINTIN , Error estimates for a semiexplicit numerical scheme for Stefan-type problems, Numer. Math., 52 (1988), 165-185. | MR 923709 | Zbl 0617.65125
[82] A. VISINTIN , Stefan problem with phase relaxation, IMA J. Appl. Math., 34 (1985), 225-245. | MR 804824 | Zbl 0585.35053
[83] A. VISINTIN , Models of Phase Transitions, Birkhäuser, Boston (1996). | MR 1423808 | Zbl 0882.35004
[84] A. VISINTIN , Nucleation and mean curvature flow, Comm. Partial Differential Equations, to appear. | MR 1608492 | Zbl 0901.53045
[85] A. VISINTIN , Introduction to the models of phase transitions, this volume, p. 1. | fulltext mini-dml | MR 1619027 | Zbl 0903.35097

La collezione può essere raggiunta anche a partire da EuDML, la biblioteca digitale matematica europea, e da mini-DML, il progetto mini-DML sviluppato e mantenuto dalla cellula Math-Doc di Grenoble.

Per suggerimenti o per segnalare eventuali errori, scrivete a

logo MBACCon il contributo del Ministero per i Beni e le Attività Culturali