Numerical Methods for Phase Transition Problems

Bollettino della Unione Matematica Italiana B

Volumn , Issue 1, 1998, Pages 83-108

  Verdi, Claudio ^a  

^a UNIVERSITY OF MILAN   (Italy)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0042295980     PISSN: 03924041     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (85)

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.
3. R. ALMGREN, Variational algorithms and pattern formation in dendritic solidification, J. Comput. Phys., 106 (1993), 337-354.
4. S. ALTSCHULER - S. B. ANGENENT - Y. GIGA, Mean curvature flow through singularities for surfaces of rotation, J. Geom. Anal., 5 (1995), 293-358.
5. I. ATHANASOPOULOS - L. CAFFARELLI - S. SALSA, Degenerate phase transition problems of parabolic type. Smoothness of the front, to appear.
6. G. BARLES - H. M. SONER - P. E. SOUGANIDIS, Front propagation and phase field theory, SIAM J. Control Optim., 31 (1993), 439-469.
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.
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.
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.
10. G. BELLETTINI - M. PAOLINI, Anisotropic motion by mean curvature in the context of Finsler geometry, Hokkaido Math. J., 25 (1996), 537-566.
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.
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.
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.
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.
15. E. BÄNSCH, Local mesh refinement in 2 and 3 dimensions, IMPACT Comput. Sci. Engrg., 3 (1991), 181-191.
16. G. CAGINALP, An analysis of a phase field model of a free boundary, Arch. Rational Mech. Anal., 92 (1986), 205-245.
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.
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.
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).
22. PH. CLÉMENT, Approximation by finite element functions using local regularization, RAIRO Modél. Math. Anal. Numér., 9 (1975), 77-84.
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.
24. G. DAL MASO, An Introduction to Γ-Convergence, Birkhäuser, Boston (1993).
25. T. DUPONT, Mesh modification for evolution equations, Math. Comp., 29 (1982), 85-107.
26. G. DZIUK, An algorithm for evolutionary surfaces, Numer. Math., 58 (1991), 603-611.
27. G. DZIUK, Convergence of a semi-discrete scheme for the curvature shortening flow, Math. Models Methods Appl. Sci., 4 (1994), 589-606.
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.
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.
31. C. M. ELLIOTT - R. SCHÄTZLE, The limit of the anisotropic double-obstacle Allen-Cahn equation, Proc. Roy. Soc. London Ser. A, to appear.
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.
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.
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.
35. L. C. EVANS - J. SPRUCK, Motion of level sets by mean curvature. I, J. Diff. Geom., 33 (1991), 635-681.
36. F. FIERRO, Numerical approximation for the mean curvature flow with nucleation using implicit time-stepping: an adaptive algorithm, Calcolo, to appear.
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.
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.
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.
40. G. HUISKEN, Flow by mean curvature of convex surfaces into spheres, J. Diff. Geom., 20 (1994), 237-266.
41. J. W. JEROME - M. E. ROSE, Error estimates for the multidimensional two-phase Stefan problem, Math. Comp., 39 (1982), 377-414.
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.
43. 0 finite element method for a model of polymer crystallization, Comput. Meth. Appl. Mech. Engrg., 125 (1995), 303-317.
44. 1 finite element method for a phase relaxation model. Part II: adaptively refined meshes, SIAM J. Numer. Anal., to appear.
45. M. KIMURA, Numerical analysis for moving boundary problems using the boundary tracking method, Japan J. Indust. Appl. Math., to appear.
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).
48. O. A. LADYZENSKAJA - V. SOLONNIKOV - N. URAL'CEVA, Linear and Quasilinear Equations of Parabolic Type, vol. TMM 23, AMS, Providence (1968).
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.
50. S. LUCKHAUS - T. STURZENHECKER, Implicit time discretization for the mean curvature flow equation, Calc. Var. Partial Differential Equations, 3 (1995), 253-271.
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.
52. R. H. NOCHETTO, Error estimates for multidimensional singular parabolic problems, Japan J. Indust. Appl. Math., 4 (1987), 111-138.
53. R. H. NOCHETTO, A stable extrapolation method for multidimensional degenerate parabolic problems, Math. Comp., 53 (1989), 455-470.
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.
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.
56. R. H. NOCHETTO - M. PAOLINI - C. VERDI, An adaptive finite elements method for two-phase Stefan problems in two space dimensions. Part II: implementation and numerical experiments, SIAM J. Sci. Statist. Comput., 12 (1991), 1207-1244.
57. R. H. NOCHETTO - M. PAOLINI - C. VERDI, A fully discrete adaptive nonlinear Chernoff formula, SIAM J. Numer. Anal., 30 (1993), 991-1014.
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.
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.
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.
61. R. H. NOCHETTO - M. PAOLINI - C. VERDI, A dynamic mesh method for curvature dependent evolving interfaces, J. Comput. Phys., 123 (1996), 296-310.
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.
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.
67. R. H. NOCHETTO - C. VERDI, Approximation of degenerate parabolic problems using numerical integration, SIAM J. Numer. Anal., 25 (1988), 784-814.
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.
69. R. H. NOCHETTO - C. VERDI, Convergence of double obstacle problems to the generalized geometric motion af fronts, SIAM J. Math. Anal., 26 (1995), 1514-1526.
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.
71. R. H. NOCHETTO - C. VERDI, Combined effect of explicit time-stepping and quadrature for curvature driven flows, Numer. Math., 74 (1996), 105-136
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.
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.
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.
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.
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.
77. A. SCHMIDT, Computation of three dimensional dendrites with finite elements, J. Comput. Phys., 125 (1996), 293-312.
78. J. A. SETHIAN, Level Set Methods, Cambridge University Press, Cambridge (1996).
79. C. VERDI, Optimal error estimates for an approximation of degenerate parabolic problems, Numer. Funct. Anal. Optim., 9 (1987), 657-670.
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.
81. C. VERDI - A. VISINTIN, Error estimates for a semiexplicit numerical scheme for Stefan-type problems, Numer. Math., 52 (1988), 165-185.
82. A. VISINTIN, Stefan problem with phase relaxation, IMA J. Appl. Math., 34 (1985), 225-245.
83. A. VISINTIN, Models of Phase Transitions, Birkhäuser, Boston (1996).
84. A. VISINTIN, Nucleation and mean curvature flow, Comm. Partial Differential Equations, to appear.
85. A. VISINTIN, Introduction to the models of phase transitions, this volume, p. 1.