Eulerian finite element method for parabolic PDEs on implicit surfaces

Interfaces and Free Boundaries

Volumn 10, Issue 1, 2008, Pages 119-138

  Dziuk, G ^a   Elliott, C M ^b  

^a UNIVERSITY OF FREIBURG   (Germany)

^b UNIVERSITY OF WARWICK   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 42449090766     PISSN: 14639963     EISSN: None     Source Type: Journal    
DOI: 10.4171/IFB/182     Document Type: Article

References (30)

1. ADALSTEINSSON, D., & SETHIAN, J. A. Transport and diffusion of material quantities on propagating interfaces via level set methods. J. Comput. Phys. 185 (2003), 271-288. Zbl 1047.76093 MR 2010161
2. AUBIN, TH., Nonlinear Analysis on Manifolds. Monge-Ampère Equations. Springer, Berlin (1982). Zbl 0512.53044 MR 0681859
3. BERTALMIO, M., CHENG, L. T., OSHER, S., & SAPIRO, G. Variational problems and partial differential equations on implicit surfaces. J. Comput. Phys. 174 (2001), 759-780. Zbl 0991.65055 MR 1868103
4. BLOWEY, J. F., & ELLIOTT, C. M. The Cahn-Hilliard gradient theory for phase separation with non-smooth free energy. Part II: Numerical Analysis. Eur. J. Appl. Math. 3 (1992), 147-179. Zbl 0810.35158 MR 1166255
5. BURGER, M., Finite element approximation of elliptic partial differential equations on implicit surfaces. Comput. Visual. Sci. (2007), to appear.
6. CHENG, L., BURCHARD, P., MERRIMAN, B., & OSHER, S. Motion of curves constrained on surfaces using a level set approach. J. Comput. Phys. 175 (2002), 604-644. Zbl 0996.65013 MR 1880120
7. COPETTI, M., & ELLIOTT, C. M. Numerical analysis of the Cahn-Hilliard equation with a logarithmic free energy. Numer. Math. 63 (1992), 39-65. Zbl 0762.65074 MR 1182511
8. DECKELNICK, K. P., DZIUK, G., ELLIOTT, C. M., & HEINE, C. J. An h-narrow band finite element method for elliptic equations on implicit surfaces. In preparation.
9. DECKELNICK, K., ELLIOTT, C. M., & STYLES, V. Numerical diffusion induced grain boundary motion. Interfaces Free Bound. 3 (2001), 393-414. Zbl 0991.35095 MR 1869586
10. DOGEL, J., TSEKOV, R., & FREYLAND, W. Two dimensional connective nano-structures of electrode-posited Zn on Au (111) induced by spinodal decomposition. J. Chemical Phys. 122 (2005), 094703, 8 pp.
11. DZIUK, G., Finite elements for the Beltrami operator on arbitrary surfaces. Partial Differential Equations and Calculus of Variations, S. Hildebrandt and R. Leis (eds.), Lecture Notes in Math. 1357, Springer, Berlin (1988), 142-155. Zbl 0663.65114 MR 0976234
12. DZIUK, G., & ELLIOTT, C. M. Finite elements on evolving surfaces. IMAJ. Numer. Anal. 27 (2007), 262-292. Zbl 1120.65102 MR 2317005
13. DZIUK, G., & ELLIOTT, C. M. Surface finite elements for parabolic equations. J. Comput. Math. 25 (2007), 385-407. MR 2337402
14. DZIUK, G., & ELLIOTT, C. M. Eulerian level set method for PDEs on evolving surfaces. Comput. Visual. Sci., to appear.
15. EILKS, C., & ELLIOTT, C. M. Numerical simulation of dealloying by surface dissolution via the evolving surface finite element method. In preparation.
16. ELLIOTT, C. M., The Cahn-Hilliard model for the kinetics of phase separation. Mathematical Models for Phase Change Problems, J. F. Rodrigues (ed.), Int. Ser. Numer. Math. 88, Birkhauser (1989), 35-73. Zbl 0692.73003 MR 1038064
17. ELLIOTT, C. M., & ZHENG, S. M. On the Cahn-Hilliard equation. Arch. Ration. Mech. Anal. 96 (1986), 339-357. Zbl 0624.35048 MR 0855754
18. GILBARG, D., & TRUDINGER, N. S. Elliptic Partial Differential Equations of Second Order. Springer (1988).
19. GREER, J. B., An improvement of a recent Eulerian method for solving PDEs on general geometries. J. Sci. Comput. 29 (2006), 321-352. Zbl 1122.65073 MR 2272322
20. GREER, J., BERTOZZI, A., & SAPIRO, G. Fourth order partial differential equations on general geometries. J. Comput. Phys. 216 (2006), 216-246. Zbl 1097.65087 MR 2223442
21. HALPERN, D., JENSEN, O. E., & GROTBERG, J. B. A theoretical study of surfactant and liquid delivery into the lung. J. Appl. Physiology 85 (1998), 333-352.
22. KIMMEL, R., Intrinsic scale space for images on surfaces: The geodesic curvature flow. Graph. Models Image Process. 59 (1997), 365-372.
23. MEMOLI, F., SAPIRO, G., & THOMPSON, P. Implicit brain imaging. Neurolmage 23 (2004), S179-S188.
24. MYERS, T. G., CHARPIN, J. P. F., & CHAPMAN, S. J. The flow and solidification of a thin fluid film on an arbitrary three dimensional surface. Phys. Fluids 14 (2002), 2788-2803. MR 1917313
25. RATZ, A., & VOIGT, A. PDEs on surfaces - a diffuse interface approach. Comm. Math. Sci. 4 (2006), 575-590. MR 2247931
26. SCHMIDT, A., & SIEBERT, K. G. Design of Adaptive Finite Element Software: The Finite Element Toolbox ALBERTA. Lecture Notes in Comput. Sci. Engrg. 42, Springer (2005). Zbl 1068.65138 MR 2127659
27. SCHONBORN, O., & DESAI, R. C. Kinetics of phase ordering on curved surfaces. Phys. A 239 (1997), 412-419.
28. SPIRA, A., & KIMMEL, R. Geometric curve flows on parametric manifolds. J. Comput. Phys. 223 (2007), 235-249. Zbl 1118.53004 MR 2314390
29. TANG, P., QIU, F., ZHANG, H. D., & YANG, Y. L. Phase separation patterns for diblock copolymers on spherical surfaces: A finite volume method. Phys. Rev. E 72 (2005), 016710, 7 pp.
30. XU, J.-J.,. & ZHAO, H.-K. An Eulerian formulation for solving partial differential equations along a moving interface. J. Sci. Comput. 19 (2003), 573-594. Zbl 1081.76579 MR 2028859