Numerical methods for phase transition problems

Claudio Verdi

Bollettino dell'Unione Matematica Italiana (1998)

  • Volume: 1-B, Issue: 1, page 83-108
  • ISSN: 0392-4041

How to cite

top

Verdi, Claudio. "Numerical methods for phase transition problems." Bollettino dell'Unione Matematica Italiana 1-B.1 (1998): 83-108. <http://eudml.org/doc/195021>.

@article{Verdi1998,
author = {Verdi, Claudio},
journal = {Bollettino dell'Unione Matematica Italiana},
keywords = {phase transition; Stefan problem; mean curvature flows},
language = {eng},
month = {2},
number = {1},
pages = {83-108},
publisher = {Unione Matematica Italiana},
title = {Numerical methods for phase transition problems},
url = {http://eudml.org/doc/195021},
volume = {1-B},
year = {1998},
}

TY - JOUR
AU - Verdi, Claudio
TI - Numerical methods for phase transition problems
JO - Bollettino dell'Unione Matematica Italiana
DA - 1998/2//
PB - Unione Matematica Italiana
VL - 1-B
IS - 1
SP - 83
EP - 108
LA - eng
KW - phase transition; Stefan problem; mean curvature flows
UR - http://eudml.org/doc/195021
ER -

References

top
  1. ALLEN, S. M.- CAHN, J. W., A macroscopic theory for antiphase boundary motion and its application to antiphase domain coarsing, Acta Metall. Mater., 27 (1979), 1085-1095. 
  2. ALMGREN, F.- TAYLOR, J. E.- WANG, L., Curvature-driven flows: a variational approach, SIAM J. Control Optim., 31 (1993), 387-438. Zbl0783.35002MR1205983
  3. ALMGREN, R., Variational algorithms and pattern formation in dendritic solidification, J. Comput. Phys., 106 (1993), 337-354. Zbl0787.65095MR1218734
  4. ALTSCHULER, S.- ANGENENT, S. B.- GIGA, Y., Mean curvature flow through singularities for surfaces of rotation, J. Geom. Anal., 5 (1995), 293-358. Zbl0847.58072MR1360824
  5. ATHANASOPOULOS, I.- CAFFARELLI, L.- SALSA, S., Degenerate phase transition problems of parabolic type. Smoothness of the front, to appear. Zbl0924.35197
  6. BARLES, G.- SONER, H. M.- SOUGANIDIS, P. E., Front propagation and phase field theory, SIAM J. Control Optim., 31 (1993), 439-469. Zbl0785.35049MR1205984
  7. BELLETTINI, G.- PAOLINI, M., 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. Zbl0826.35051MR1298266
  8. BELLETTINI, G.- PAOLINI, M., Quasi-optimal error estimates for the mean curvature flow with a forcing term, Diff. Integ. Eq., 8 (1995), 735-752. Zbl0820.49019MR1306590
  9. BELLETTINI, G.- PAOLINI, M., 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. Zbl0944.53039MR1387549
  10. BELLETTINI, G.- PAOLINI, M., Anisotropic motion by mean curvature in the context of Finsler geometry, Hokkaido Math. J., 25 (1996), 537-566. Zbl0873.53011MR1416006
  11. BELLETTINI, G.- PAOLINI, M.- VERDI, C., 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. Zbl0754.65065MR1118855
  12. BELLETTINI, G.- PAOLINI, M.- VERDI, C., Convex approximations of functionals with curvature, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. (9) Mat. Appl., 2 (1991), 297-306. Zbl0754.65066MR1152636
  13. BLOWEY, J. F.- ELLIOTT, C. M., 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. Zbl0809.35168MR1277388
  14. BREZZI, F.- CAFFARELLI, L. A., Convergence of the discrete free boundaries for finite element approximations, RAIRO Modél. Math. Anal. Numér., 17 (1983), 385-395. Zbl0547.65081MR713766
  15. BÄNSCH, E., Local mesh refinement in 2 and 3 dimensions, IMPACT Comput. Sci. Engrg., 3 (1991), 181-191. Zbl0744.65074MR1141298
  16. CAGINALP, G., An analysis of a phase field model of a free boundary, Arch. Rational Mech. Anal., 92 (1986), 205-245. Zbl0608.35080MR816623
  17. CHEN, X.- REITICH, F., Local existence and uniqueness of solutions of the Stefan problem with surface tension and kinetic undercooling, J. Math. Anal. Appl., 164 (1992), 350-362. Zbl0761.35113MR1151039
  18. CHEN, Y. G.- GIGA, Y.- GOTO, S., Uniqueness and existence of viscosity solutions of generalized mean curvature flow equation, J. Diff. Geom., 33 (1991), 749-786. Zbl0696.35087MR1100211
  19. CHEN, Z.- NOCHETTO, R. H., A posteriori error estimation for the continuous casting problem, in preparation. 
  20. CHEN, Z.- NOCHETTO, R. H., A posteriori error estimation and adaptivity for phase relaxation models, in preparation. 
  21. CIARLET, P. G., The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam (1978). Zbl0383.65058MR520174
  22. CLÉMENT, H., Approximation by finite element functions using local regularization, RAIRO Modél. Math. Anal. Numér., 9 (1975), 77-84. Zbl0368.65008MR400739
  23. CRANDALL, M. G.- ISHII, H.- LIONS, P. L., User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1-67. Zbl0755.35015MR1118699
  24. DAL MASO, G., An Introduction to G-Convergence, Birkhäuser, Boston (1993). Zbl0816.49001MR1201152
  25. DUPONT, T., Mesh modification for evolution equations, Math. Comp., 29 (1982), 85-107. Zbl0493.65044MR658215
  26. DZIUK, G., An algorithm for evolutionary surfaces, Numer. Math., 58 (1991), 603-611. Zbl0714.65092MR1083523
  27. DZIUK, G., Convergence of a semi-discrete scheme for the curvature shortening flow, Math. Models Methods Appl. Sci., 4 (1994), 589-606. Zbl0811.65112MR1291140
  28. DZIUK, G., Convergence of a semi-discrete scheme for the anisotropic curvature shortening flow, to appear. 
  29. ELLIOTT, C. M., Error analysis of the enthalpy method for the Stefan problem, IMA J. Numer. Anal., 7 (1987), 61-71. Zbl0638.65088MR967835
  30. ELLIOTT, C. M.- PAOLINI, M.- SCHÄTZLE, R., Interface estimates for the fully anisotropic Allen-Cahn equation and anisotropic mean curvature flow, Math. Models Methods Appl. Sci., 6 (1996), 1103-1118. Zbl0873.35039MR1428147
  31. ELLIOTT, C. M.- SCHÄTZLE, R., The limit of the anisotropic double-obstacle Allen-Cahn equation, Proc. Roy. Soc. London Ser. A, to appear. Zbl0865.35073MR1424223
  32. ELLIOTT, C. M.- SCHÄTZLE, R., The limit of the fully anisotropic double-obstacle Allen-Cahn equation in the non-smooth case, SIAM J. Math. Anal., to appear. Zbl0870.35128MR1434036
  33. ERIKSSON, K.- JOHNSON, C., Adaptive finite element methods for parabolic problems I: a linear model problem, SIAM J. Numer. Anal., 28 (1991), 43-77. Zbl0732.65093MR1083324
  34. EVANS, L. C.- SONER, H. M.- SOUGANIDIS, P. E., Phase transitions and generalized motion by mean curvature, Comm. Pure Appl. Math., 45 (1992), 1097-1123. Zbl0801.35045MR1177477
  35. EVANS, L. C.- SPRUCK, J., Motion of level sets by mean curvature. I, J. Diff. Geom., 33 (1991), 635-681. Zbl0726.53029MR1100206
  36. FIERRO, F., Numerical approximation for the mean curvature flow with nucleation using implicit time-stepping: an adaptive algorithm, Calcolo, to appear. Zbl0927.65144MR1740750
  37. FIERRO, F.- GOGLIONE, R.- PAOLINI, M., Numerical simulations of mean curvature flow in presence of a nonconvex anisotropy, Math. Models Methods Appl. Sci., to appear. Zbl0946.58014MR1634826
  38. FIFE, P. C.- PENROSE, O., Thermodynamically consistent models of phase-field type for the kinetics of phase transitions, Physica D, 43 (1990), 44-62. Zbl0709.76001MR1060043
  39. GIGA, Y.- GOTO, S.- ISHII, H.- SATO, M. H., Comparison principle and convexity preserving properties for singular degenerate parabolic equations on unbounded domains, Indiana Univ. Math. J., 40 (1991), 443-470. Zbl0836.35009MR1119185
  40. HUISKEN, G., Flow by mean curvature of convex surfaces into spheres, J. Diff. Geom., 20 (1994), 237-266. Zbl0556.53001MR772132
  41. JEROME, J. W.- ROSE, M. E., Error estimates for the multidimensional two-phase Stefan problem, Math. Comp., 39 (1982), 377-414. Zbl0505.65060MR669635
  42. JIANG, X.- NOCHETTO, R. H., A finite element method for a phase relaxation model. Part I: quasi-uniform mesh, SIAM J. Numer. Anal., to appear. Zbl0972.65067MR1619875
  43. JIANG, X.- NOCHETTO, R. H.- VERDI, C., A P 12P0 finite element method for a model of polymer crystallization, Comput. Meth. Appl. Mech. Engrg., 125 (1995), 303-317. Zbl0949.82025MR1352100
  44. JIANG, X.- NOCHETTO, R. H.- VERDI, C., A P 12P1 finite element method for a phase relaxation model. Part II: adaptively refined meshes, SIAM J. Numer. Anal., to appear. Zbl0934.65105MR1688994
  45. KIMURA, M., Numerical analysis for moving boundary problems using the boundary tracking method, Japan J. Indust. Appl. Math., to appear. Zbl0892.76065MR1475140
  46. KLEIN, O. W., Existence and approximation results for phase-field systems of Penrose-Fife type and Stefan problems, Ph.D. Thesis, Humboldt-Universität, Berlin (1997). 
  47. KORNHUBER, R., Adaptive Monotone Multigrid Methods for Nonlinear Variational Problems, TeubnerStuttgart (1997). Zbl0879.65041MR1469497
  48. LADYZENSKAJA, O. A.- SOLONNIKOV, V.- URAL'CEVA, N., Linear and Quasilinear Equations of Parabolic Type, vol. TMM 23, AMS, Providence (1968). Zbl0174.15403MR241822
  49. LUCKHAUS, S., Solutions for the two-phase Stefan problem with the Gibbs-Thomson law for the melting temperature, European J. Appl. Math., 1 (1990), 101-111. Zbl0734.35159MR1117346
  50. LUCKHAUS, S.- STURZENHECKER, T., Implicit time discretization for the mean curvature flow equation, Calc. Var. Partial Differential Equations, 3 (1995), 253-271. Zbl0821.35003MR1386964
  51. MAGENES, E.- NOCHETTO, R. H.- VERDI, C., Energy error estimates for a linear scheme to approximate nonlinear parabolic problems, RAIRO Modél. Math. Anal. Numér., 21 (1987), 655-678. Zbl0635.65123MR921832
  52. NOCHETTO, R. H., Error estimates for multidimensional singular parabolic problems, Japan J. Indust. Appl. Math., 4 (1987), 111-138. Zbl0657.65132MR899207
  53. NOCHETTO, R. H., A stable extrapolation method for multidimensional degenerate parabolic problems, Math. Comp., 53 (1989), 455-470. Zbl0675.65112MR982372
  54. NOCHETTO, R. H., 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. Zbl0733.65089MR1138471
  55. NOCHETTO, R. H.- PAOLINI, M.- VERDI, C., 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. Zbl0733.65087MR1079028
  56. NOCHETTO, R. H.- PAOLINI, M.- VERDI, C., 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. Zbl0733.65088MR1114983
  57. NOCHETTO, R. H.- PAOLINI, M.- VERDI, C., A fully discrete adaptive nonlinear Chernoff formula, SIAM J. Numer. Anal., 30 (1993), 991-1014. Zbl0805.65135MR1231324
  58. NOCHETTO, R. H.- PAOLINI, M.- VERDI, C., Sharp error analysis for curvature dependent evolving fronts, Math. Models Methods Appl. Sci., 3 (1993), 711-723. Zbl0802.65124MR1245632
  59. NOCHETTO, R. H.- PAOLINI, M.- VERDI, C., Optimal interface error estimates for the mean curvature flow, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 21 (1994), 193-212. Zbl0886.35079MR1288364
  60. NOCHETTO, R. H.- PAOLINI, M.- VERDI, C., Double obstacle formulation with variable relaxation parameter for smooth geometric front evolutions: asymptotic interface error estimates, Asymptotic Anal., 10 (1995), 173-198. Zbl0852.35060MR1324387
  61. NOCHETTO, R. H.- PAOLINI, M.- VERDI, C., A dynamic mesh method for curvature dependent evolving interfaces, J. Comput. Phys., 123 (1996), 296-310. Zbl0851.65067MR1372375
  62. NOCHETTO, R. H.- PAOLINI, M.- VERDI, C., Numerical Analysis of Geometric Motion of Fronts, CRM, Montreal, in preparation. 
  63. NOCHETTO, R. H.- SCHMIDT, A.- VERDI, C., A posteriori error estimation and adaptivity for degenerate parabolic problems, Math. Comp., to appear. Zbl0942.65111MR1648399
  64. NOCHETTO, R. H.- SCHMIDT, A.- VERDI, C., Mesh and time step modification for degenerate parabolic problems, in preparation. 
  65. NOCHETTO, R. H.- SCHMIDT, A.- VERDI, C., Adaptive algorithm and simulations for Stefan problems in two and three dimensions, in preparation. 
  66. NOCHETTO, R. H.- SCHMIDT, A.- VERDI, C., Adapting meshes and time-steps for phase change problems, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. (9) Mat. Appl., to appear. Zbl0910.65106MR1631617
  67. NOCHETTO, R. H.- VERDI, C., Approximation of degenerate parabolic problems using numerical integration, SIAM J. Numer. Anal., 25 (1988), 784-814. Zbl0655.65131MR954786
  68. NOCHETTO, R. H.- VERDI, C., An efficient linear scheme to approximate parabolic free boundary problems: error estimates and implementation, Math. Comp., 51 (1988), 27-53. Zbl0657.65131MR942142
  69. NOCHETTO, R. H.- VERDI, C., Convergence of double obstacle problems to the generalized geometric motion of fronts, SIAM J. Math. Anal., 26 (1995), 1514-1526. Zbl0839.35008MR1356457
  70. NOCHETTO, R. H.- VERDI, C., 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. Zbl0844.76007MR1365307
  71. NOCHETTO, R. H.- VERDI, C., Combined effect of explicit time-stepping and quadrature for curvature driven flows, Numer. Math., 74 (1996), 105-136 Zbl0859.65066MR1400218
  72. NOCHETTO, R. H.- VERDI, C., Convergence past singularities for a fully discrete approximation of curvature driven interfaces, SIAM J. Numer. Anal., 34 (1997), 490-512. Zbl0876.35053MR1442924
  73. OSHER, S.- SETHIAN, J. A., Fronts propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., 79 (1988), 12-49. Zbl0659.65132MR965860
  74. PAOLINI, M., 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. Zbl0838.73079MR1365309
  75. PAOLINI, M.- VERDI, C., Asymptotic and numerical analyses of the mean curvature flow with a space-dependent relaxation parameter, Asymptotic Anal., 5 (1992), 553-574. Zbl0757.65078MR1169358
  76. RULLA, J.- WALKINGTON, N. J., Optimal rates of convergence for degenerate parabolic problems in two dimensions, SIAM J. Numer. Anal., 33 (1996), 56-67. Zbl0856.65102MR1377243
  77. SCHMIDT, A., Computation of three dimensional dendrites with finite elements, J. Comput. Phys., 125 (1996), 293-312. Zbl0844.65096
  78. SETHIAN, J. A., Level Set Methods, Cambridge University Press, Cambridge (1996). Zbl0859.76004MR1409367
  79. VERDI, C., Optimal error estimates for an approximation of degenerate parabolic problems, Numer. Funct. Anal. Optim., 9 (1987), 657-670. Zbl0598.65091MR895990
  80. VERDI, C., 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. Zbl0819.35155MR1321834
  81. VERDI, C.- VISINTIN, A., Error estimates for a semiexplicit numerical scheme for Stefan-type problems, Numer. Math., 52 (1988), 165-185. Zbl0617.65125MR923709
  82. VISINTIN, A., Stefan problem with phase relaxation, IMA J. Appl. Math., 34 (1985), 225-245. Zbl0585.35053MR804824
  83. VISINTIN, A., Models of Phase Transitions, Birkhäuser, Boston (1996). Zbl0882.35004MR1423808
  84. VISINTIN, A., Nucleation and mean curvature flow, Comm. Partial Differential Equations, to appear. Zbl0901.53045MR1608492
  85. VISINTIN, A., Introduction to the models of phase transitions, this volume, p. 1. Zbl0903.35097MR1619027

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.