A robust solver for a second order mixed finite element method for the Cahn-Hilliard equation. (English) Zbl 1478.65080
The authors propose a second-order-in-time mixed finite element scheme for the Cahn-Hilliard equation. The idea is to split the Cahn-Hilliard equation into two coupled second order PDEs and use mixed finite elements in spatial discretization, while the second order convex splitting approach is employed in temporal discretization to achieve the unconditional energy stability. A Newton iteration based solver was developed to solve the fully discrete nonlinear system at each time step. The key ingredient of the solver is a preconditioned minimal residual algorithm (with a multigrid preconditioner) whose performance is independent of the spatial mesh size and the time step size for a given interfacial width parameter. Numerical experiments were shown to confirm the performance of the proposed second order scheme, which showed its advantage over the first order version previously developed by the authors [J. Sci. Comput. 77, No. 2, 1234–1249 (2018; Zbl 1407.65179)].
Reviewer: Yongyong Cai (Beijing)
MSC:
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 65M55 | Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs |
| 65F08 | Preconditioners for iterative methods |
| 65F10 | Iterative numerical methods for linear systems |
| 35Q35 | PDEs in connection with fluid mechanics |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65L06 | Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations |
Keywords:
Cahn-Hilliard equation; convex splitting; mixed finite element methods; MINRES; block diagonal preconditioner; multigridCitations:
Zbl 1407.65179References:
| [1] | Brenner, S. C.; Diegel, A. E.; Sung, L.-Y., A robust solver for a mixed finite element method for the Cahn-Hilliard equation, J. Sci. Comput., 77, 1234 (2018) · Zbl 1407.65179 |
| [2] | Cahn, J. W., On spinodal decomposition, Acta Metall., 9, 795 (1961) |
| [3] | Temam, R., Infinite-Dimensional Dynamical Systems in Mechanics and Physics (1988), Springer-Verlag: Springer-Verlag New York · Zbl 0662.35001 |
| [4] | Cahn, J. W.; Hilliard, J. E., Free energy of a nonuniform system. I. interfacial free energy, J. Chem. Phys., 28, 258 (1958) · Zbl 1431.35066 |
| [5] | Elliott, C. M.; Zheng, S., On the Cahn-Hilliard equation, Arch. Ration. Mech. Anal., 96, 339-357 (1986) · Zbl 0624.35048 |
| [6] | Cai, Y.; Shen, J., Error estimates for a fully discretized scheme to a Cahn-Hilliard phase-field model for two-phase incompressible flows, Math. Comp., 87, 313, 2057-2090 (2018) · Zbl 1390.35225 |
| [7] | Chen, Y.; Lowengrub, J.; Shen, J.; Wang, C.; Wise, S. M., Efficient energy stable schemes for isotropic and strongly anisotropic Cahn-Hilliard systems with the Willmore regularization, J. Comput. Phys., 365, 56-73 (2018) · Zbl 1395.65014 |
| [8] | Choksi, R.; Maras, M.; Williams, J. F., 2d phase diagram for minimizers of a Cahn-Hilliard functional with long-range interactions, SIAM J. Appl. Dyn. Syst., 10, 4, 1344-1362 (2011) · Zbl 1236.49074 |
| [9] | Feng, X., Fully discrete finite element approximations of the Navier-Stokes-Cahn-Hilliard diffuse interface model for two-phase fluid flows, SIAM J. Numer. Anal., 44, 1049-1072 (2006) · Zbl 1344.76052 |
| [10] | Lee, H. G.; Lowengrub, J. S.; Goodman, J., Modeling pinchoff and reconnection in a Hele-Shaw cell. I. the models and their calibration, Phys. Fluids, 14, 492-513 (2002) · Zbl 1184.76316 |
| [11] | van Teeffelen, S.; Backofen, R.; Voigt, A.; Löwen, H., Derivation of the phase-field-crystal model for colloidal solidification, Phys. Rev. E, 79, Article 051404 pp. (2009) |
| [12] | Ainsworth, M.; Mao, Z., Analysis and approximation of a fractional Cahn-Hilliard equation, SIAM J. Numer. Anal., 55, 4, 1689-1718 (2017) · Zbl 1369.65124 |
| [13] | Feng, W.; Guan, Z.; Lowengrub, J.; Wang, C.; Wise, S. M., A uniquely solvable, energy stable numerical scheme for the functionalized Cahn-Hilliard equation and its convergence analysis, J. Sci. Comput., 1-30 (2018) |
| [14] | Furihata, D.; Kovács, M.; Larsson, S.; Lindgren, F., Strong convergence of a fully discrete finite element approximation of the stochastic Cahn-Hilliard equation, SIAM J. Numer. Anal., 56, 2, 708-731 (2018) · Zbl 1390.60231 |
| [15] | Song, H.; Shu, C. W., Unconditional energy stability analysis of a second order implicit-explicit local discontinuous Galerkin method for the Cahn-Hilliard equation, J. Sci. Comput., 73, 2-3, 1178-1203 (2017) · Zbl 1383.65105 |
| [16] | Yang, X.; Zhao, J.; Wang, Q.; Shen, J., Numerical approximations for a three-component Cahn-Hilliard phase-field model based on the invariant energy quadratization method, Math. Models Methods Appl. Sci., 27, 11, 1993-2030 (2017) · Zbl 1393.80003 |
| [17] | Wang, J.; Zhai, Q.; Zhang, R.; Zhang, S., A weak Galerkin finite element scheme for the Cahn-Hilliard equation, Math. Comp., 88, 211-235 (2019) · Zbl 1420.65128 |
| [18] | Aristotelous, A.; Karakasian, O.; Wise, S. M., Adaptive, second-order in time, primitive-variable discontinuous Galerkin schemes for a Cahn-Hilliard equation with a mass source, IMA J. Numer. Anal., 35, 1167-1198 (2015) · Zbl 1323.65106 |
| [19] | Guillén-González, F.; Tierra, G., Second order schemes and time-step adaptivity for Allen-Cahn and Cahn-Hilliard models, Comput. Math. Appl., 68, 8, 821-846 (2014) · Zbl 1362.65104 |
| [20] | Guo, J.; Wang, C.; Wise, S. M.; Yue, X., An \(H^2\) convergence of a second-order convex-splitting, finite difference scheme for the three-dimensional Cahn-Hilliard equation, Commun. Math. Sci., 14, 489-515 (2016) · Zbl 1338.65221 |
| [21] | Han, D.; Brylev, A.; Yang, X.; Tan, Z., Numerical analysis of second order, fully discrete energy stable schemes for phase field models of two phase incompressible flows, J. Sci. Phys., 70, 965-989 (2017) · Zbl 1397.76070 |
| [22] | Liu, F.; Shen, J., Stabilized semi-implicit spectral deferred correction methods for Allen-Cahn and Cahn-Hilliard equations, Math. Methods Appl. Sci., 38, 18, 4564-4575 (2015) · Zbl 1335.65078 |
| [23] | Yan, Y.; Chen, W.; Wang, C.; Wise, S. M., A second order energy stable BDF numerical scheme for the Cahn-Hilliard equation, Commun. Comput. Phys., 23, 572-602 (2018) · Zbl 1488.65367 |
| [24] | Zhou, J.; Chen, L.; Huang, Y.; Wang, W., An efficient two-grid scheme for the Cahn-Hilliard equation, Commun. Comput. Phys., 17, 1, 127-145 (2015) · Zbl 1373.76111 |
| [25] | Diegel, A.; Wang, C.; Wise, S. M., Stability and convergence of a second-order mixed finite element method for the Cahn-Hilliard equation, IMA J. Numer. Anal., 36, 1867-1897 (2016) · Zbl 1433.80005 |
| [26] | Eyre, D., Unconditionally gradient stable time marching the Cahn-Hilliard equation, MRS Online Proc. Library Arch., 529 (1998) |
| [27] | Aristotelous, A.; Karakasian, O.; Wise, S. M., A mixed discontinuous Galerkin, convex splitting scheme for a modified Cahn-Hilliard equation and an efficient nonlinear multigrid solver, Discrete Contin. Dyn. Syst. Ser. B, 18, 9 (2013) · Zbl 1278.65149 |
| [28] | Feng, W.; Salgado, A.; Wang, C.; Wise, S. M., Preconditioned steepest descent methods for some nonlinear elliptic equations involving p-Laplacian terms, J. Comput. Phys., 334, 45-67 (2017) · Zbl 1375.35149 |
| [29] | Guo, R.; Xu, Y., Efficient solvers of discontinuous Galerkin discretizations for the Cahn-Hilliard equations, J. Sci. Comput., 58, 2, 380-408 (2014) · Zbl 1296.65134 |
| [30] | Kay, D.; Welford, R., A multigrid finite element solver for the Cahn-Hilliard equation, J. Comput. Phys., 212, 1, 288-304 (2006) · Zbl 1081.65091 |
| [31] | Shin, J.; Kim, S.; Lee, D.; Kim, J., A parallel multigrid method of the Cahn-Hilliard equation, Comput. Mater. Sci., 71, 89-96 (2013) |
| [32] | Bänsch, E.; Morin, P.; Nochetto, R. H., Preconditioning a class of fourth order problems by operator splitting, Numer. Math., 118, 197-228 (2011) · Zbl 1241.65031 |
| [33] | Wise, S. M.; Kim, J.; Lowengrub, J., Solving the regularized, strongly anisotropic Cahn-Hilliard equation by an adaptive nonlinear multigrid method, J. Comput. Phys., 226, 414-446 (2007) · Zbl 1310.82044 |
| [34] | Zheng, B.; Chen, L.-P.; Hu, X.; Chen, L.; Nochetto, R. H.; Xu, J., Fast multilevel solvers for a class of discrete fourth order parabolic problems, J. Sci. Comput., 69, 201-226 (2016) · Zbl 1410.65386 |
| [35] | Brenner, S. C.; Scott, L. R., The Mathematical Theory of Finite Element Methods (2008), Springer-Verlag: Springer-Verlag New York · Zbl 1135.65042 |
| [36] | Hu, Z.; Wise, S. M.; Wang, C.; Lowengrub, J. S., Stable and efficient finite-difference nonlinear-multigrid schemes for the phase-field crystal equation, J. Comput. Phys., 228, 5323-5339 (2009) · Zbl 1171.82015 |
| [37] | Zulehner, W., Nonstandard norms and robust estimates for saddle point problems, SIAM J. Matrix Anal. Appl., 32, 2, 536-560 (2011) · Zbl 1251.65078 |
| [38] | Greenbaum, A., Iterative Methods for Solving Linear Systems (1997), SIAM: SIAM Philadelphia · Zbl 0883.65022 |
| [39] | Elman, H. C.; Silvester, D. J.; Wathen, A. J., Finite Elements and Fast Iterative Solvers: With Applications in Incompressible Fluid Dynamics (2014), Oxford University Press: Oxford University Press Oxford · Zbl 1304.76002 |
| [40] | Benzi, M.; Golub, G. H.; Liesen, J., Numerical solution of saddle point problems, Acta Numer., 14, 1-137 (2005) · Zbl 1115.65034 |
| [41] | Mardal, K. A.; Winther, R., Preconditioning discretizations of systems of partial differential equations, Numer. Linear Algebra Appl., 18, 1-40 (2011) · Zbl 1249.65246 |
| [42] | Walker, S. W., FELICITY: A matlab/c++ toolbox for developing finite element methods and simulation modeling, SIAM J. Sci. Comput., 40, 2, C234-C257 (2018) · Zbl 1481.65192 |
| [43] | Hackbusch, W., Multi-Grid Methods and Applications (1985), Springer-Verlag: Springer-Verlag Berlin-Heidelberg-New York-Tokyo · Zbl 0585.65030 |
| [44] | Trottenberg, U.; Oosterlee, C.; Schüller, A., Multigrid (2001), Academic Press: Academic Press San Diego · Zbl 0976.65106 |
| [45] | Diegel, A.; Feng, X.; Wise, S. M., Analysis of a mixed finite element method for a Cahn-Hilliard-Darcy-Stokes system, SIAM J. Numer. Anal., 53, 1, 127-152 (2015) · Zbl 1330.76065 |
| [46] | Diegel, A.; Wang, C.; Wang, X.; Wise, S. M., Convergence analysis and error estimates for a second order accurate finite element method for the Cahn-Hilliard-Navier-Stokes system, Numer. Math., 137, 495 (2017) · Zbl 1523.65081 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
