Mathematics of Computation

Fully discrete dynamic mesh discontinuous Galerkin methods for the Cahn-Hilliard equation of phase transition

Author(s): Xiaobing Feng; Ohannes A. Karakashian.
Journal: Math. Comp. 76 (2007), 1093-1117.
MSC (2000): Primary 65M15, 65M60, 74N20
Posted: March 9, 2007
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Abstract: Fully discrete discontinuous Galerkin methods with variable mesh- es in time are developed for the fourth order Cahn-Hilliard equation arising from phase transition in materials science. The methods are formulated and analyzed in both two and three dimensions, and are proved to give optimal order error bounds. This coupled with the flexibility of the methods demonstrates that the proposed discontinuous Galerkin methods indeed provide an efficient and viable alternative to the mixed finite element methods and nonconforming (plate) finite element methods for solving fourth order partial differential equations.

1.: R. Adams Sobolev Spaces Academic Press, New York, 1975. MR 0450957 (56:9247)
2.: N. D. Alikakos, P. W. Bates, and X. Chen, Convergence of the Cahn-Hilliard equation to the Hele-Shaw model, Arch. Rational Mech. Anal., (128), 1994, pp. 165-205. MR 1308851 (97b:35174)
3.: D. Arnold, An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal., (19), 1982, pp. 742-760. MR 664882 (83f:65173)
4.: D. Arnold, F. Brezzi, B. Cockburn, and D. Marini, Discontinuous Galerkin methods for elliptic problems. Proceedings of the International Symposium on the discontinuous Galerkin method (B. Cockburn, G.E. Karniadakis, C.-W. Shu Eds.), Springer Lecture Notes in Computational Science and Engineering, vol. 11, 2000, pp. 89-101.
5.: D. Arnold, F. Brezzi, B. Cockburn, and D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal., (39), 2001, pp. 1749-1779. MR 1885715 (2002k:65183)
6.: G. A. Baker, Finite element methods for elliptic equations using nonconforming elements. Math. Comp., (31), 1977, pp. 44-59. MR 0431742 (55:4737)
7.: G. A. Baker, W. N. Jureidini and O. A. Karakashian, Piecewise solenoidal vector fields and the Stokes problems. SIAM J. Numer. Anal., (27), 1990, pp. 1466-1485. MR 1080332 (91m:65246)
8.: J. H. Bramble, T. Dupont, and V. Thomée, Projection methods for Dirichlet's problem in approximating polygonal domains with boundary-value corrections, Math. Comp., (26), 1972, pp.869-879. MR 0343657 (49:8397)
9.: S. Brenner and R. Scott, The Mathematical Theory of Finite Element Methods, Springer-Verlag, New York, 1994. MR 1278258 (95f:65001)
10.: J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system I. Interfacial free energy, J. Chem. Phys., (28), 1958, pp. 258-267.
11.: P. G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam, 1978. MR 0520174 (58:25001)
12.: B. Cockburn, G. E. Karniadakis, C.-W. Shu, Discontinuous Galerkin Methods, Theory, Computation, and Applications, Springer Lecture Notes in Computational Science and Engineering, vol. 11, Springer-Verlag, 2000. MR 1842160 (2002b:65004)
13.: B Cockburn and C. -W. Shu, The local discontinuous Galerkin method for convection-diffusion systems, SIAM J. Numer. Anal., (35), 1998, pp. 2440-2463. MR 1655854 (99j:65163)
14.: J. Douglas, Jr. and T. Dupont, Galerkin methods for parabolic problems, SIAM J. Numer. Anal., (7), 1970, pp. 575-626. MR 0277126 (43:2863)
15.: J. Douglas, Jr. and T. Dupont, Interior penalty procedures for elliptic and parabolic Galerkin methods, Lecture Notes In Physics 58, Springer Verlag, Berlin, 1976. MR 0440955 (55:13823)
16.: Q. Du and R. A. Nicolaides, Numerical analysis of a continuum model of phase transition, SIAM J. Numer. Anal., (28), 1991, pp. 1310-1322. MR 1119272 (92h:65166)
17.: C. M. Elliott and D. A. French, A nonconforming finite-element method for the two-dimensional Cahn-Hilliard equation, SIAM J. Numer. Anal., (26), 1989, pp. 884-903. MR 1005515 (90k:65163)
18.: C. M. Elliott, D. A. French, and F. A. Miller, A second order splitting method for the Cahn-Hilliard equation, Numer. Math. (54), 1989, pp. 575-590. MR 978609 (90k:65173)
19.: C. M. Elliott and S. Zheng, On the Cahn-Hilliard equation, Arch. Rational Mech. Anal., (96), 1986, pp. 339-357. MR 855754 (87k:80007)
20.: X. Feng, Fully discrete finite element approximations of the Navier-Stokes-Cahn-Hilliard diffuse interface model for two-phase fluid flows, SIAM J. Numer. Anal. (44), 2006, pp. 1049-1072. MR 2231855
21.: X. Feng and O. A. Karakashian, Two-level nonoverlapping additive Schwarz methods for a discontinuous Galerkin approximation of the biharmonic problem, J. Scient. Comput. (22), 2005, pp. 299-324. MR 2142199 (2006b:65156)
22.: X. Feng and A. Prohl, Error Analysis of a mixed finite element method for the Cahn-Hilliard equation, Numer. Math. (99), 2004, pp. 47-84. MR 2101784 (2005m:65191)
23.: X. Feng and A. Prohl, Numerical analysis of the Cahn-Hilliard equation and approximation for the Hele-Shaw problem, Interfaces and Free Boundaries, (7), 2005, pp. 1-28. MR 2126141 (2005j:35098)
24.: X. Feng and H. Wu, A posteriori error estimates and an adaptive finite element algorithm for the Cahn-Hilliard equation and the Hele-Shaw flow, M2AN Math. Model. Numer. Anal. (submitted).
25.: D. A. French and S. Jensen, Long-time behaviour of arbitrary order continuous time Galerkin schemes for some one-dimensional phase transition problems, IMA J. Numer. Anal., (14), 1994, pp. 421-442. MR 1283945 (95d:65083)
26.: P. Grisvard, Singularities in Boundary Value Problems, Research Notes in Applied Mathematics, Vol. 22, MASSON and Springer-Verlag, 1992. MR 1173209 (93h:35004)
27.: P. Houston, E. Süli and C. Schwab, Discontinuous -finite element methods for advection-diffusion problems, SIAM J. Numer. Anal. (39), 2002, pp. 2133-2163. MR 1897953 (2003d:65108)
28.: D. Jacqmin, Calculation of two-phase Navier-Stokes flows using phase-field modeling, J. Comp. Phys., (115), 1999, pp. 96-127. MR 1716497 (2000g:76088)
29.: O. A. Karakashian and W. N. Jureidini, A nonconforming finite element method for the stationary Navier-Stokes Equations. SIAM J. Numer. Anal., (35), 1998, pp. 93-120. MR 1618436 (99d:65320)
30.: O. A. Karakashian, C. Makridakis, Convergence of a continuous Galerkin method with mesh modification for nonlinear wave equations, Math. Comp. (74), 2005, pp. 85-102 MR 2085403 (2005g:65147)
31.: C. Liu and J. Shen, A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method, Physica D, (179), 2003, pp. 211-228. MR 1984386 (2004j:35233)
32.: G. B. McFadden, Phase field models of solidification, Contemporary Mathematics, (295), 2002, pp. 107-145. MR 1940624 (2003k:80004)
33.: I. Mozolevski and E. Süli, A priori error analysis for the -version of the discontinuous Galerkin finite element method for the biharmonic equation, Comput. Meth. Appl. Math. (3), 2003, pp. 596-607. MR 2048235 (2005c:65106)
34.: J.T. Oden and C.E. Baumann, A conservative DGM for convection-diffusion and Navier-Stokes problems, Proceedings of the International Symposium on the discontinuous Galerkin method (B. Cockburn, G. E. Karniadakis, C.-W. Shu Eds.), Springer Lecture Notes in Computational Science and Engineering, vol. 11, 2000, pp. 179-196. MR 1842173 (2002d:65128)
35.: M. F. Wheeler, A priori error estimates for Galerkin approximations to parabolic partial differential equations, SIAM J. Numer. Anal., (10), 1973, pp. 723-759. MR 0351124 (50:3613)
36.: M. F. Wheeler, An elliptic collocation-finite element method with interior penalties, SIAM J. Numer. Anal., (15), 1978, pp. 152-161 MR 0471383 (57:11117)

Retrieve articles in Mathematics of Computation with MSC (2000): 65M15, 65M60, 74N20

Xiaobing Feng
Affiliation: Department of Mathematics, The University of Tennessee, Knoxville, Tennessee 37996
Email: xfeng@math.utk.edu

Ohannes A. Karakashian
Affiliation: Department of Mathematics, The University of Tennessee, Knoxville, Tennessee 37996
Email: ohannes@math.utk.edu

DOI: 10.1090/S0025-5718-07-01985-0
PII: S 0025-5718(07)01985-0
Keywords: Biharmonic equation, Cahn-Hilliard equation, discontinuous Galerkin methods, dynamic meshes, error estimates
Received by editor(s): February 1, 2006
Received by editor(s) in revised form: August 31, 2006
Posted: March 9, 2007
Additional Notes: The work of the first author was partially supported by the NSF grant DMS-0410266
The work of the second author was partially supported by the NSF grant DMS-0411448
Copyright of article: Copyright 2007, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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