Analysis of a diffuse interface approach to an advection diffusion equation on a moving surface

Mathematical Models and Methods in Applied Sciences

Volumn 19, Issue 5, 2009, Pages 787-802

  Elliott, Charles M ^a   Stinner, Björn ^a  

^a UNIVERSITY OF WARWICK   (United Kingdom)

Author keywords

Asymptotic analysis; Moving surface; Surface partial diferential equation; Weighted Sobolev space

Indexed keywords

ADVECTION DIFFUSION EQUATION; COEFFICIENT FUNCTIONS; CONVERGENCE OF SOLUTIONS; DIFFUSE INTERFACE; DIFFUSE INTERFACE MODELS; ENERGY METHOD; EXISTENCE AND UNIQUENESS; INTERFACIAL LAYER; LIQUID-LIQUID INTERFACES; MOVING SURFACE; MOVING SURFACES; PHASE FIELD MODELS; SPECIES DIFFUSION; SURFACE MOTIONS; WEAK SOLUTION; WEIGHTED SOBOLEV SPACE; WEIGHTED SOBOLEV SPACES;

ADVECTION; ASYMPTOTIC ANALYSIS; DIFFUSION; EQUATIONS OF MOTION; GRAIN BOUNDARIES; LIQUIDS; PHASE INTERFACES; SURFACE ACTIVE AGENTS; SURFACE DIFFUSION; TURBULENT FLOW;

SURFACE ANALYSIS;

EID: 67650083559     PISSN: 02182025     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218202509003620     Document Type: Article

References (27)

1. D. Adalsteinsson and J. A. Sethian, Transport and diffusion of material quantities on propagating interfaces via level set methods, J. Comp. Phys. 185 (2003) 271-288.
2. F. Antoci, Some necessary and sufficient conditions for the compactness of the embedding of weighted Sobolev spaces, Ric. di Mat. LII (2003) 55-71.
3. M. Bertalmío, L.-T. Cheng, S. Osher and G. Sapiro, Variational problems and partial differential equations on implicit surfaces, J. Comp. Phys. 174 (2001) 759-780.
4. J. F. Blowey and C. M. Elliott, The Cahn-Hilliard gradient theory for phase separation with nonsmooth free energy part I: Mathematical analysis, Eur. J. Appl. Math. 2 (1991) 233-280.
5. J. F. Blowey and C. M. Elliott, Curvature Dependent Phase Boundary Motion and Parabolic Obstacle Problems, in Degenerate Diffusion, eds. W.-M. Ni, L. A. Peletier and J. L. Vasquez, IMA, Vol. 47, Math. Appl., Vol. 47 (Springer-Verlag, 1993), pp. 19-60.
6. G. Caginalp, Stefan and Hele-Shaw type models as asymptotic limits of the phase field equations, Phys. Rev. A 39 (1989) 5887-5896.
7. K. Deckelnick, G. Dziuk, C. M. Elliott and C.-J. Heine, An h-narrow band finite element method for implicit surfaces, IMA J. Numer. Anal., to appear.
8. K. Deckelnick, C. M. Elliott and V. Styles, Numerical diffusion-induced grain boundary motion, Interf. Free Bound. 3 (2001) 393-414.
9. G. Dziuk and C. M. Elliott, Finite elements on evolving surfaces, IMA J. Numer. Anal. 25 (2007) 385-407.
10. G. Dziuk and C. M. Elliott, Eulerian finite element method for parabolic pdes on implicit surfaces, Interf. Free Bound. 10 (2008) 119-138.
11. G. Dziuk and C. M. Elliott, An Eulerian approach to transport and diffusion on evolving implicit surfaces, Comput. Visual Sci. online First, doi: 10.1007/s00791-008-0122-0.
12. C. Eilks and C. M. Elliott, Numerical simulation of dealloying by surface dissolution via the evolving surface finite element method, J. Comput. Phys. 227 (2008) 9727-9741.
13. P. C. Fife, J. W. Cahn and C. M. Elliott, A free-boundary model for diffusion-induced grain boundary motion, Interf. Free Bound. 3 (2001) 291-336.
14. J. B. Greer, An improvement of a recent Eulerian method for solving PDEs on general geometries, J. Sci. Comput. 29 (2006) 321-352.
15. J. B. Greer, A. L. Bertozzi and G. Sapiro, Fourth order partial differential equations on general geometries, J. Comp. Phys. 216 (2006) 216-246.
16. J. Hale and G. Raugel, Reaction-diffusion equations on thin domains, J. Math. Pures Appl. 71 (1992) 33-95.
17. J. K. Hale and G. Raugel, Partial differential equations on thin domains, in Differential Equations and Mathematical Physics, Birmingham, AL, 1990, Math. Sci. Engrg. Vol. 186 (Academic Press, 1992), pp. 63-97.
18. A. J. James and J. Lowengrub, A surfactant-conserving volume-of-fluid method for interfacial flows with insoluble surfactant, J. Comp. Phys. 201 (2004) 685-722.
19. J. Lowengrub, J.-J. Xu and A. Voigt, Surface phase separation and flow in a simple model of multicomponent drops and vesicles, Fluid Dyn. Mat. Proc. 3 (2007) 1-20.
20. U. F. Mayer and G. Simonett, Classical solutions for diffusion-induced grain-boundary motion, J. Math. Anal. Appl. 234 (1999) 660-674.
21. M. Prizzi, M. Rinaldi and K. P. Rybakowski, Curved thin domains and parabolic equations, Stud. Math. 151 (2002) 109-140.
22. A. Rätz and A. Voigt, Pde's on surfaces - a diffuse interface approach, Comm. Math. Sci. 4 (2006) 575-590.
23. A. Rätz and A. Voigt, A diffuse-interface approximation for surface diffusion including adatoms, Nonlin. 20 (2007) 177-192.
24. J. Rodríguez and J. M. Viaño, Asymptotic analysis of Poisson's equation in a thin domain and its application to thin-walled elastic beams and tubes, Math. Meth. Appl. Sci. 21 (1998) 187-226.
25. P. Schwartz, D. Adalsteinsson, P. Colella, A. P. Arkin and M. Onsum, Numerical computation of diffusion on a surface, Proc. Natl. Acad. Sci. 102 (2005) 11151 - 11156.
26. J. Simon, Compact sets in the space Lp(0, T; B), Ann. Mat. Pura Appl. 146 (1986) 65-96.
27. J.-J. Xu and H.-K. Zhao, An Eulerian formulation for solving partial differential equations along a moving interface, J. Sci. Comp. 19 (2003) 573-594.