×

zbMATH — the first resource for mathematics

On fully practical finite element approximations of degenerate Cahn-Hilliard systems. (English) Zbl 0987.35071
The authors have developed fully practical finite element approximations of degenerate Cahn-Hillard systems with a degenerate mobility matrix. A number of theorems and convergence analysis has been made for uniqueness of solution, and its foundation is established by a number of experiments. It is quite an interesting paper for them who are working in the area of finite element techniques.

MSC:
35K35 Initial-boundary value problems for higher-order parabolic equations
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35K65 Degenerate parabolic equations
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35K55 Nonlinear parabolic equations
82C26 Dynamic and nonequilibrium phase transitions (general) in statistical mechanics
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML
References:
[1] R.A. Adams and J. Fournier , Cone conditions and properties of Sobolev spaces . J. Math. Anal. Appl. 61 ( 1977 ) 713 - 734 . Zbl 0385.46024 · Zbl 0385.46024 · doi:10.1016/0022-247X(77)90173-1
[2] J.W. Barrett and J.F. Blowey , An error bound for the finite element approximation of a model for phase separation of a multi-component alloy . IMA J. Numer. Anal. 16 ( 1996 ) 257 - 287 . Zbl 0849.65069 · Zbl 0849.65069 · doi:10.1093/imanum/16.2.257
[3] J.W. Barrett and J.F. Blowey , Finite element approximation of a model for phase separation of a multi-component alloy with non-smooth free energy . Numer. Math. 77 ( 1997 ) 1 - 34 . Zbl 0882.65129 · Zbl 0882.65129 · doi:10.1007/s002110050276
[4] J.W. Barrett and J.F. Blowey , Finite element approximation of a model for phase separation of a multi-component alloy with a concentration dependent mobility matrix . IMA J. Numer. Anal. 18 ( 1998 ) 287 - 328 . Zbl 0899.76262 · Zbl 0899.76262 · doi:10.1093/imanum/18.2.287
[5] J.W. Barrett and J.F. Blowey , Finite element approximation of a model for phase separation of a multi-component alloy with non-smooth free energy and a concentration dependent mobility matrix . M\(^{\,3}\)AS 9 ( 1999 ) 627 - 663 . Zbl 0936.65120 · Zbl 0936.65120 · doi:10.1142/S0218202599000336
[6] J.W. Barrett and J.F. Blowey , An improved error bound for a finite element approximation of a model for phase separation of a multi-component alloy with a concentration dependent mobility matrix . Numer. Math. 88 ( 2001 ) 255 - 297 . Zbl 0990.65105 · Zbl 0990.65105 · doi:10.1007/s002110000205
[7] J.W. Barrett , J.F. Blowey and H. Garcke , Finite element approximation of a fourth order nonlinear degenerate parabolic equation . Numer. Math. 80 ( 1998 ) 525 - 556 . Zbl 0913.65084 · Zbl 0913.65084 · doi:10.1007/s002110050377
[8] J.W. Barrett , J.F. Blowey and H. Garcke , Finite element approximation of the Cahn-Hilliard equation with degenerate mobility . SIAM J. Numer. Anal. 37 ( 1999 ) 286 - 318 . Zbl 0947.65109 · Zbl 0947.65109 · doi:10.1137/S0036142997331669
[9] J.F. Blowey , M.I.M. Copetti and C.M. Elliott , The numerical analysis of a model for phase separation of a multi-component alloy . IMA J. Numer. Anal. 16 ( 1996 ) 111 - 139 . Zbl 0857.65137 · Zbl 0857.65137 · doi:10.1093/imanum/16.1.111
[10] J.F. Blowey and C.M. Elliott , The Cahn-Hilliard gradient theory for phase separation with non-smooth free energy, part i: Mathematical analysis . European J. Appl. Math. 2 ( 1991 ) 233 - 279 . Zbl 0797.35172 · Zbl 0797.35172 · doi:10.1017/S095679250000053X
[11] J.F. Blowey and C.M. Elliott , The Cahn-Hilliard gradient theory for phase separation with non-smooth free energy, part ii: Numerical analysis . European J. Appl. Math. 3 ( 1992 ) 147 - 179 . Zbl 0810.35158 · Zbl 0810.35158 · doi:10.1017/S0956792500000759
[12] L. Bronsard , H. Garcke and B. Stoth , A multi-phase Mullins-Sekerka system: matched asymptotic expansions and an implicit time discretisation for the geometric evolution problem , in Proc. Roy. Soc. Edinburgh 128 A ( 1998 ) 481 - 506 . Zbl 0924.35199 · Zbl 0924.35199 · doi:10.1017/S0308210500021612
[13] J.F. Cialvaldini , Analyse numérique d’un problème de Stefan à deux phases par une méthode d’éléments finis . SIAM J. Numer. Anal. 12 ( 1975 ) 464 - 487 . Zbl 0272.65101 · Zbl 0272.65101 · doi:10.1137/0712037
[14] P.G. Ciarlet , Introduction to numerical linear algebra and optimisation . C.U.P., Cambridge ( 1988 ). MR 1015713 | Zbl 0672.65001 · Zbl 0672.65001
[15] D. de Fontaine , An analysis of clustering and ordering in multicomponent solid solutions - I . Stability criteria. J. Phys. Chem. Solids 33 ( 1972 ) 297 - 310 .
[16] P.G. de Gennes , Dynamics of fluctuations and spinodal decomposition in polymer blends . J. Chem. Phys. 72 ( 1980 ) 4756 - 4763 . Zbl 1110.82310 · Zbl 1110.82310 · doi:10.1063/1.439809
[17] C.M. Elliott , The Cahn-Hilliard model for the kinetics of phase transitions , in Mathematical models for phase change problems, J.F. Rodrigues Ed., Internat. Ser. Numer. Math. 88, Birkhäuser-Verlag, Basel ( 1989 ) 35 - 73 . Zbl 0692.73003 · Zbl 0692.73003
[18] C.M. Elliott and H. Garcke , On the Cahn-Hilliard equation with degenerate mobility . SIAM J. Math. Anal. 27 ( 1996 ) 404 - 423 . Zbl 0856.35071 · Zbl 0856.35071 · doi:10.1137/S0036141094267662
[19] C.M. Elliott and H. Garcke , Diffusional phase transitions in multicomponent systems with a concentration dependent mobility matrix . Physica D 109 ( 1997 ) 242 - 256 . Zbl 0925.35087 · Zbl 0925.35087 · doi:10.1016/S0167-2789(97)00066-3
[20] C.M. Elliott and S. Luckhaus , A generalized diffusion equation for phase separation of a multi-component mixture with interfacial free energy . SFB256 University Bonn, Preprint 195 ( 1991 ).
[21] D.J. Eyre , Systems of Cahn-Hilliard equations . SIAM J. Appl. Math. 53 ( 1993 ) 1686 - 1712 . Zbl 0853.73060 · Zbl 0853.73060 · doi:10.1137/0153078
[22] H. Garcke , B. Nestler and B. Stoth , Anisotropy in multi phase systems: a phase field approach . Interfaces Free Bound. 1 ( 1999 ) 175 - 198 . Zbl 0959.35169 · Zbl 0959.35169 · doi:10.4171/IFB/8
[23] H. Garcke and A. Novick-Cohen , A singular limit for a system of degenerate Cahn-Hilliard equations . Adv. Diff. Eq. 5 ( 2000 ) 401 - 434 . Zbl 0988.35019 · Zbl 0988.35019
[24] G. Grün and M. Rumpf , Nonnegativity preserving numerical schemes for the thin film equation . Numer. Math. 87 ( 2000 ) 113 - 152 . Zbl 0988.76056 · Zbl 0988.76056 · doi:10.1007/s002110000197
[25] K. Ito and Y. Kohsaka , Three-phase boundary motion by surface diffusion: stability of a mirror symmetric stationary solution . Interfaces Free Bound. 3 ( 2001 ) 45 - 80 . Zbl 0972.35165 · Zbl 0972.35165 · doi:10.4171/IFB/32
[26] P.L. Lions and B. Mercier , Splitting algorithms for the sum of two nonlinear operators . SIAM J. Numer. Anal. 16 ( 1979 ) 964 - 979 . Zbl 0426.65050 · Zbl 0426.65050 · doi:10.1137/0716071
[27] J.E. Morral and J.W. Cahn , Spinodal decomposition in ternary systems . Acta Metall. 19 ( 1971 ) 1037 - 1045 .
[28] A. Novick-Cohen , The Cahn-Hilliard equation: mathematical and modelling perspectives . Adv. Math. Sci. Appl. 8 ( 1998 ) 965 - 985 . Zbl 0917.35044 · Zbl 0917.35044
[29] F. Otto and W. E, Thermodynamically driven incompressible fluid mixtures. J. Chem. Phys. 107 (1997) 10177-10184. Zbl 1110.76307 · Zbl 1110.76307
[30] L. Zhornitskaya and A.L. Bertozzi , Positivity preserving numerical schemes for lubrication-type equations . SIAM J. Numer. Anal. 37 ( 2000 ) 523 - 555 . Zbl 0961.76060 · Zbl 0961.76060 · doi:10.1137/S0036142998335698
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.