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## High order finite element calculations for the deterministic Cahn-Hilliard equation

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203 | Free energy of a nonuniform system. i. interfacial free energy - Cahn, Hilliard - 1958 |

131 |
The p-version of the finite element method
- Babuška, Szabo, et al.
- 1981
(Show Context)
Citation Context ...bifurcation occurs as 1 > ρ1. ε2 Below, we study this first bifurcation and illustrate theoretical results of [23]. 4.1 Asymptotic stable states on a rectangle In the case of a rectangular domain Ω = =-=[0,2]-=-×[0,1], the hypothesis of the Theorem 4.1 in [23] holds. Accordingly, if 1 ε2 > ρ1 := π2 4 then there exist exactly two attractors ±uε which can be expressed as ±uε(x,y) = ± 2ε √ 1 π2 √ − 3 ε2 4 cos (... |

129 |
On spinodal decomposition
- Cahn
- 1961
(Show Context)
Citation Context ...del for the evolution of an isotropic system of nonuniform composition or density. They introduced a free energy density ¯ f to define a chemical potential, anduseitintheclassicaltransportequation(see=-=[10]-=-, [12]and[13]). Thetotalfreeenergy F of the binary alloy is a volume integral on Ω of this free energy density (bulk free energy): ∫ F := ¯f(u,∇u,∇ 2 u,...) dV. (0.1) Ω They assumed ¯ f to be a functi... |

121 |
Finite element solution of the Helmholtz equation with high wave number. II. The h-p version of the FEM
- Ihlenburg, Babuška
- 1997
(Show Context)
Citation Context ...odal finite elements under a constant mobility M ≡ 1. The use of p-version (increasing polynomial degree, see [2]) instead of h-version (decrease mesh-step) has proved to be efficient for propagation =-=[1, 19, 20]-=-, corner singularities [34], or oscillating problems [9]. The numerical results obtained here with the finiteelementlibraryMélina[27]showthatthismethod issuitableintheCahn-Hilliardframework as well. O... |

112 |
Front migration in the nonlinear Cahn-Hilliard equation
- Pego
- 1989
(Show Context)
Citation Context ... interfacial energies. These diffuse interfaces are shortened in an effect resembling the surface tension on a sharp interface, as the material fronts move to reduce their own curvature (see [11] and =-=[31]-=-). Finally, the solution reaches an equilibrium the location and form of which depend on the total initial concentration (see [24]). Nevertheless this equilibrium is always a solution with an interfac... |

70 |
Spectral methods
- Bernardi, Maday
- 1997
(Show Context)
Citation Context ...o 64 (the nodes may be chosen as the Gauss-Lobatto points to avoid Runge phenomenon for large degrees). It can thus be used as a p-version code – see [2] – or even to implement spectral methods – see =-=[6]-=-. In the following results, we use quadrangular elements for two-dimensional computations,withdegreefrom1to10. SoweusethenotationQi withi ∈ {1,2,3,4,5,6,7,8,9,10} to describe these elements. We justif... |

68 |
The Cahn-Hilliard gradient theory for phase separation with nonsmooth free energy
- Blowey, Elliott
- 1991
(Show Context)
Citation Context ..., 17, 25, 26, 32, 33, 37]. Recently, Ma and Wang have studied the stationary solutions of the Cahn-Hiliard equation (see [23]). The case of non smooth ψ has been the object of much less research (see =-=[8]-=- and [15]). Other frequent simplifications are often made. The mobility M is often assumed to be constant and the physical parameters are set to 1 - as we have done above in (0.9). For a more physical... |

60 | Finite element approximation of the Cahn-Hilliard equation with degenerate mobility
- Barrett, Blowey, et al.
- 1999
(Show Context)
Citation Context ...or (related to ψ), and with (w0,u0) the finite dimensional representation of the initial data. The system (1.4) is clearly block-symmetric. The proof of the convergence of this scheme can be found in =-=[3]-=-. hal-00461231, version 1 - 4 Mar 2010 1.2 Nonlinear solver At each time step, we use a Newton procedure to solve the implicit nonlinear system (1.4) . For (1.4), we define the operator L by: ( τA M L... |

56 | The Cahn-Hilliard equation: mathematical and modeling perspectives - Novick-Cohen - 1998 |

54 |
Nonlinear aspects of the Cahn-Hilliard equation
- Novick-Cohen, Segel
- 1984
(Show Context)
Citation Context ...erivative: f : u ↦→ 1 ( 2 1−u 4 ) 2 , (0.9) ψ : u ↦→ u 3 −u. (0.10) The Cahn-Hilliard equation has been extensively studied in the case where ψ is replaced by a polynomialfunction (see [12], [22] and =-=[30]-=-). Furthermore, this model hasbeen used successfully for describing phase separationphenomena, see for example the survey[28], and the referencestherein, or other recent results on spinodal decomposit... |

53 | The dynamics of nucleation for the Cahn-Hilliard equation
- Bates, Fife
- 1993
(Show Context)
Citation Context ...odel hasbeen used successfully for describing phase separationphenomena, see for example the survey[28], and the referencestherein, or other recent results on spinodal decomposition and nucleation in =-=[5, 7, 17, 25, 26, 32, 33, 37]-=-. Recently, Ma and Wang have studied the stationary solutions of the Cahn-Hiliard equation (see [23]). The case of non smooth ψ has been the object of much less research (see [8] and [15]). Other freq... |

49 |
The Cahn-Hilliard equation with a concentration dependent mobility: motion by minus the Laplacian of the mean curvature
- Cahn, Elliott, et al.
- 1996
(Show Context)
Citation Context ...uce their interfacial energies. These diffuse interfaces are shortened in an effect resembling the surface tension on a sharp interface, as the material fronts move to reduce their own curvature (see =-=[11]-=- and [31]). Finally, the solution reaches an equilibrium the location and form of which depend on the total initial concentration (see [24]). Nevertheless this equilibrium is always a solution with an... |

46 |
Linking anisotropic sharp and diffuse surface motion laws via gradient flows
- Taylor, Cahn
- 1994
(Show Context)
Citation Context ...are often made. The mobility M is often assumed to be constant and the physical parameters are set to 1 - as we have done above in (0.9). For a more physically relevant choice of mobility, we mention =-=[36]-=- where the following form is proposed M(u) = max{0,1−u 2 }. Among the physical parameters, ε has a peculiar role since it may lead to different asymptotic behaviors and equilibria (see [24] and sectio... |

45 |
Discrete dispersion relation for hp-version finite element approximation at high wave number
- Ainsworth
(Show Context)
Citation Context ...odal finite elements under a constant mobility M ≡ 1. The use of p-version (increasing polynomial degree, see [2]) instead of h-version (decrease mesh-step) has proved to be efficient for propagation =-=[1, 19, 20]-=-, corner singularities [34], or oscillating problems [9]. The numerical results obtained here with the finiteelementlibraryMélina[27]showthatthismethod issuitableintheCahn-Hilliardframework as well. O... |

34 |
Numerical studies of the Cahn-Hilliard equation for phase separation
- Elliott, French
- 1987
(Show Context)
Citation Context ...vative of f4 := u ↦→ Tc ( 1−u 2 2 ) [ ] 2 u u4 +T + +K4. (2.5) 2 12 Then a stationary solution of the Cahn-Hilliard equation (0.6) can be explicitly computed (under a constant mobility M(u) ≡ 1), see =-=[16]-=-: where u+ = √ 3 u : x ↦→ u+tanh(xµ), (2.6) ( Tc T −1 ) √ Tc −T and µ = ε √ . (2.7) 2 It is important to remark that the solution is constrained in [−u+,u+]. We can define a characteristic length ℓ (s... |

31 | Spinodal decomposition for the Cahn-Hilliard equation in higher dimensions: nonlinear dynamics
- Maier-Paape, Wanner
- 2000
(Show Context)
Citation Context ...odel hasbeen used successfully for describing phase separationphenomena, see for example the survey[28], and the referencestherein, or other recent results on spinodal decomposition and nucleation in =-=[5, 7, 17, 25, 26, 32, 33, 37]-=-. Recently, Ma and Wang have studied the stationary solutions of the Cahn-Hiliard equation (see [23]). The case of non smooth ψ has been the object of much less research (see [8] and [15]). Other freq... |

31 | Counting stationary solutions of the Cahn-Hilliard equation by transversality arguments - Grinfeld, Novick-Cohen - 1995 |

27 |
Theory of spinodal decomposition in alloys
- Langer
- 1971
(Show Context)
Citation Context ...m: with derivative: f : u ↦→ 1 ( 2 1−u 4 ) 2 , (0.9) ψ : u ↦→ u 3 −u. (0.10) The Cahn-Hilliard equation has been extensively studied in the case where ψ is replaced by a polynomialfunction (see [12], =-=[22]-=- and [30]). Furthermore, this model hasbeen used successfully for describing phase separationphenomena, see for example the survey[28], and the referencestherein, or other recent results on spinodal d... |

26 |
Numerical analysis of the Cahn-Hilliard equation with a logarithmic free energy
- Copetti, Elliott
- 1992
(Show Context)
Citation Context ...on stays far from ±1 so that no special care is needed. This is expected. Indeed, it is known that in the one dimensional case the solution satisfies an L ∞ bound which is strictly less than one (see =-=[14]-=-). The same result has not been proved in higher dimension but it is probably true. A simple remark shows the mass conservation through the total scheme. Indeed, if we multiply the first component of ... |

23 | Spinodal decomposition for the Cahn-Hilliard equation
- Grant
- 1993
(Show Context)
Citation Context ...odel hasbeen used successfully for describing phase separationphenomena, see for example the survey[28], and the referencestherein, or other recent results on spinodal decomposition and nucleation in =-=[5, 7, 17, 25, 26, 32, 33, 37]-=-. Recently, Ma and Wang have studied the stationary solutions of the Cahn-Hiliard equation (see [23]). The case of non smooth ψ has been the object of much less research (see [8] and [15]). Other freq... |

22 |
A multigrid finite element solver for the Cahn-Hilliard equation
- Kay, Welford
(Show Context)
Citation Context ...daptative mesh cannot be used. In the literature, many technical ideas have been studied: adaptive refinement of the time-stepping or of the mesh, 2C 1 elements (see [35]), multigrid resolution (see =-=[21]-=-). We propose here an alternative method using high degree C 0 lagrangian nodal finite elements under a constant mobility M ≡ 1. The use of p-version (increasing polynomial degree, see [2]) instead of... |

21 |
On the Cahn-Hilliard equation with a logarithmic free energy
- Debussche, Dettori
- 1995
(Show Context)
Citation Context ..., 26, 32, 33, 37]. Recently, Ma and Wang have studied the stationary solutions of the Cahn-Hiliard equation (see [23]). The case of non smooth ψ has been the object of much less research (see [8] and =-=[15]-=-). Other frequent simplifications are often made. The mobility M is often assumed to be constant and the physical parameters are set to 1 - as we have done above in (0.9). For a more physically releva... |

21 | Unexpectedly linear behavior for the Cahn-Hilliard equation - Sander, Wanner |

16 |
Spinodal decomposition: a reprise
- Cahn, Hilliard
- 1971
(Show Context)
Citation Context ...volution of an isotropic system of nonuniform composition or density. They introduced a free energy density ¯ f to define a chemical potential, anduseitintheclassicaltransportequation(see[10], [12]and=-=[13]-=-). Thetotalfreeenergy F of the binary alloy is a volume integral on Ω of this free energy density (bulk free energy): ∫ F := ¯f(u,∇u,∇ 2 u,...) dV. (0.1) Ω They assumed ¯ f to be a function of u and i... |

16 |
Steady states of the one-dimensional Cahn-Hilliard equation
- Novick-Cohen, Peletier
- 1993
(Show Context)
Citation Context ... elements because it seems that they provide better results on the interface length as shown on Figure 18. 4 Stationary states The Cahn-Hilliard equation has a lot of asymptotic equilibria (see [18], =-=[29]-=- and [30]). In the one dimensional case, a state can be described by the number of interfaces and their positions. On Figure 23, we show four states which are numerically stable. It is possible to obs... |

16 | Monte Carlo simulations for spinodal decomposition
- Sander, Wanner
- 1999
(Show Context)
Citation Context |

14 |
Second phase spinodal decomposition for the Cahn-Hilliard-Cook equation
- Blömker, Maier-Paape, et al.
(Show Context)
Citation Context |

14 |
Maximum norms of random sums and transient pattern formation
- Wanner
(Show Context)
Citation Context |

14 | Computations of the first eigenpairs for the Schrödinger operator with magnetic field - Bonnaillie-Noël, Dauge, et al. |

11 |
On the convergence of the p-version of the boundary element Galerkin method
- Stephan, Suri
- 1989
(Show Context)
Citation Context ...ant mobility M ≡ 1. The use of p-version (increasing polynomial degree, see [2]) instead of h-version (decrease mesh-step) has proved to be efficient for propagation [1, 19, 20], corner singularities =-=[34]-=-, or oscillating problems [9]. The numerical results obtained here with the finiteelementlibraryMélina[27]showthatthismethod issuitableintheCahn-Hilliardframework as well. Our paper is organized as fo... |

5 |
The finite element library Mélina
- Martin
- 2008
(Show Context)
Citation Context ...rease mesh-step) has proved to be efficient for propagation [1, 19, 20], corner singularities [34], or oscillating problems [9]. The numerical results obtained here with the finiteelementlibraryMélina=-=[27]-=-showthatthismethod issuitableintheCahn-Hilliardframework as well. Our paper is organized as follows: in section 1, we shortly describe the discretization (in both time and space) including the nonline... |

5 | Approximation of Cahn-Hilliard diffuse interface models using parallel adaptive mesh refinement and coarsening with C1 elements - Stogner, Carey, et al. |

4 | Cahn-Hilliard equations and phase transition dynamics for binary systems
- Ma, Wang
(Show Context)
Citation Context ...rein, or other recent results on spinodal decomposition and nucleation in [5, 7, 17, 25, 26, 32, 33, 37]. Recently, Ma and Wang have studied the stationary solutions of the Cahn-Hiliard equation (see =-=[23]-=-). The case of non smooth ψ has been the object of much less research (see [8] and [15]). Other frequent simplifications are often made. The mobility M is often assumed to be constant and the physical... |

4 |
Path-following the equilibria of the Cahn-Hilliard equation on the square
- Maier-Paape, Miller
(Show Context)
Citation Context ...this difference is of the same order as each term. As in the case of the square, we can find numerically stable states corresponding to the next modes in the nomenclature of Maier-Paape and Miller in =-=[24]-=-. We have found 4 numerically stable states (see Figure 35), with energies that have been drawn on Figure 36 against the length of the interface. We can clear see the linear dependance between the two... |

1 |
Finite element approximationofthe Cahn-Hilliard equation with degenerate mobility
- Barrett, Blowey, et al.
- 1999
(Show Context)
Citation Context ...th ε → 0 is of great importance: in particular a constant mobility leads to a Mullins-Sekerka evolution (nonlocal coupling) whereas a degenerate mobility leads to a purely local geometric motion (see =-=[4]-=-). Furthermore, when the interface thickness is of the order of a nanometer, an artificially large parameter ε is often used to regularize the numerical problem. When a fine resolution is out of reach... |

1 |
Computations ofthe first eigenpairsfor the Schrödinger operator with magnetic field
- Bonnaillie-Noël, Dauge, et al.
(Show Context)
Citation Context ... p-version (increasing polynomial degree, see [2]) instead of h-version (decrease mesh-step) has proved to be efficient for propagation [1, 19, 20], corner singularities [34], or oscillating problems =-=[9]-=-. The numerical results obtained here with the finiteelementlibraryMélina[27]showthatthismethod issuitableintheCahn-Hilliardframework as well. Our paper is organized as follows: in section 1, we short... |

1 |
CountingstationarysolutionsoftheCahn-Hilliardequation by transversality arguments
- Novick-Cohen
- 1995
(Show Context)
Citation Context ...fer Q3 elements because it seems that they provide better results on the interface length as shown on Figure 18. 4 Stationary states The Cahn-Hilliard equation has a lot of asymptotic equilibria (see =-=[18]-=-, [29] and [30]). In the one dimensional case, a state can be described by the number of interfaces and their positions. On Figure 23, we show four states which are numerically stable. It is possible ... |

1 |
The Cahn-Hilliardequation: mathematicaland modelingperspectives
- Novick-Cohen
- 1998
(Show Context)
Citation Context ...n the case where ψ is replaced by a polynomialfunction (see [12], [22] and [30]). Furthermore, this model hasbeen used successfully for describing phase separationphenomena, see for example the survey=-=[28]-=-, and the referencestherein, or other recent results on spinodal decomposition and nucleation in [5, 7, 17, 25, 26, 32, 33, 37]. Recently, Ma and Wang have studied the stationary solutions of the Cahn... |

1 |
ApproximationofCahn-Hilliarddiffuseinterface models using parallel adaptive mesh refinement and coarsening with C 1 elements
- Stogner, Murray
(Show Context)
Citation Context .... When a fine resolution is out of reach, a change in the height of the barrier between wells in the free energy density, coupled with a change on ε, allows simulations with larger length scales (see =-=[35]-=- for details). The evolution of the solution of (0.6) can essentially be split into two stages. The first one is the spinodal decomposition described in section 2 where the two species quickly separat... |