zbMATH — the first resource for mathematics

A multigrid method for the Cahn-Hilliard equation with obstacle potential. (English) Zbl 1168.65386
Summary: We present a multigrid finite element method for the deep quench obstacle Cahn-Hilliard equation. The non-smooth nature of this highly nonlinear fourth order partial differential equation make this problem particularly challenging. The method exhibits mesh-independent convergence properties in practice for arbitrary time step sizes. In addition, numerical evidence shows that this behaviour extends to small values of the interfacial parameter \(\gamma \). Several numerical examples are given, including comparisons with existing alternative solution methods for the Cahn-Hilliard equation.

65M55 Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35Q53 KdV equations (Korteweg-de Vries equations)
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
Full Text: DOI
[1] Alikakos, N.D.; Bates, P.W.; Chen, X.F., The convergence of solutions of the cahn – hilliard equation to the solution of the hele – shaw model, Arch. rational mech. anal., 128, 165-205, (1994) · Zbl 0828.35105
[2] Baňas, L’.; Nürnberg, R., Finite element approximation of a three dimensional phase field model for void electromigration, J. sci. comp., 37, 2, 202-232, (2008) · Zbl 1203.65175
[3] L’. Baňas , R. Nürnberg, Phase field computations for surface diffusion and void electromigration in \(\mathbb{R}^3\), Comput. Vis. Sci., in press, doi:10.1007/s00791-008-0114-0.
[4] Barrett, J.W.; Blowey, J.F.; Garcke, H., Finite element approximation of the cahn – hilliard equation with degenerate mobility, SIAM J. numer. anal., 37, 286-318, (1999) · Zbl 0947.65109
[5] Barrett, J.W.; Nürnberg, Robert; Styles, Vanessa, Finite element approximation of a phase field model for void electromigration, SIAM J. numer. anal., 42, 2, 738-772, (2004) · Zbl 1076.78012
[6] Blowey, J.F.; Elliott, C.M., The cahn – hilliard gradient theory for phase separation with non-smooth free energy. part II: numerical analysis, European J. appl. math., 3, 147-179, (1992) · Zbl 0810.35158
[7] Cahn, J.W., On spinodal decomposition, Acta metall., 9, 795-801, (1961)
[8] Cahn, J.W.; Elliott, C.M.; Novick-Cohen, A., The cahn – hilliard equation with a concentration dependent mobility: motion by minus the Laplacian of the Mean curvature, European J. appl. math., 7, 3, 287-301, (1996) · Zbl 0861.35039
[9] Cahn, J.W.; Hilliard, J.E., Free energy of a non-uniform system. I. interfacial free energy, J. chem. phys., 28, 258-267, (1958)
[10] Feng, X.; Prohl, Andreas, Numerical analysis of the cahn – hilliard equation and approximation of the hele – shaw problem, Interfaces free bound., 7, 1, 1-28, (2005) · Zbl 1072.35150
[11] Gräser, C.; Kornhuber, R., Multigrid methods for obstacle problems, J. comput. math., 27, 1, 1-44, (2009) · Zbl 1199.65401
[12] Gräser, C.; Kornhuber, Ralf, On preconditioned Uzawa-type iterations for a saddle point problem with inequality constraints, (), 91-102
[13] Haber, E.; Heldmann, Stefan, An octree multigrid method for quasi-static maxwell’s equations with highly discontinuous coefficients, J. comput. phys., 223, 2, 783-796, (2007) · Zbl 1125.78013
[14] Hackbusch, W.; Mittelmann, H.-D., On multigrid methods for variational inequalities, Numer. math., 42, 1, 65-76, (1983) · Zbl 0497.65042
[15] Hackbusch, W., Multigrid methods and applications, Springer series in computational mathematics, vol. 4, (1985), Springer-Verlag Berlin
[16] Hoppe, R.H.W., Multigrid algorithms for variational inequalities, SIAM J. numer. anal., 24, 5, 1046-1065, (1987) · Zbl 0628.65046
[17] Imoro, B., Discretized obstacle problems with penalties on nested grids, Appl. numer. math., 32, 1, 21-34, (2000) · Zbl 0942.65072
[18] Kay, D.; Welford, Richard, A multigrid finite element solver for the cahn – hilliard equation, J. comput. phys., 212, 1, 288-304, (2006) · Zbl 1081.65091
[19] Kim, J.; Kang, Kyungkeun; Lowengrub, John, Conservative multigrid methods for cahn – hilliard fluids, J. comput. phys., 193, 2, 511-543, (2004) · Zbl 1109.76348
[20] Kornhuber, R., Monotone multigrid methods for elliptic variational inequalities I, Numer. math., 69, 167-184, (1994) · Zbl 0817.65051
[21] Mandel, J., A multilevel iterative method for symmetric, positive definite linear complementarity problems, Appl. math. optim., 11, 1, 77-95, (1984) · Zbl 0539.65046
[22] Manservisi, S., Numerical analysis of vanka-type solvers for steady Stokes and navier – stokes flows, SIAM J. numer. anal., 44, 5, 2025-2056, (2006) · Zbl 1120.76042
[23] Modica, L., Gradient theory of phase transitions with boundary contact energy, Ann. inst. H. Poincaré anal. non linéaire, 4, 487-512, (1987) · Zbl 0642.49009
[24] Schmidt, A.; Siebert, K.G., Design of adaptive finite element software: the finite element toolbox ALBERTA, Lecture notes in computational science and engineering, vol. 42, (2005), Springer-Verlag Berlin
[25] Schöberl, J.; Zulehner, Walter, On Schwarz-type smoothers for saddle point problems, Numer. math., 95, 2, 377-399, (2003) · Zbl 1033.65105
[26] Trottenberg, U.; Oosterlee, C.W.; Schüller, A., Multigrid, (2001), Academic Press Inc. San Diego, CA, With contributions by A. Brandt, P. Oswald, and K. Stüben
[27] Zhang, Y., Multilevel projection algorithm for solving obstacle problems, Comput. math. appl., 41, 12, 1505-1513, (2001) · Zbl 0985.65076
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.