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Finite element approximation of the CahnHilliard equation with degenerate mobility
 Math. Comp
, 1999
"... Abstract. We consider the CahnHilliard equation with a logarithmic free energy and nondegenerate concentration dependent mobility. In particular we prove that there exists a unique solution for sufficiently smooth initial data. Further, we prove an error bound for a fully practical piecewise linea ..."
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Cited by 31 (8 self)
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Abstract. We consider the CahnHilliard equation with a logarithmic free energy and nondegenerate concentration dependent mobility. In particular we prove that there exists a unique solution for sufficiently smooth initial data. Further, we prove an error bound for a fully practical piecewise linear finite element approximation in one and two space dimensions. Finally some numerical experiments are presented. 1.
A Phase Field Model for Continuous Clustering on Vector Fields
 IEEE Transactions on Visualization and Computer Graphics
"... A new method for the simplification of flow fields is presented. It is based on continuous clustering. A wellknown physical clustering model, the Cahn Hilliard model which describes phase separation, is modified to reflect the properties of the data to be visualized. Clusters are defined implicitly ..."
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Cited by 17 (3 self)
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A new method for the simplification of flow fields is presented. It is based on continuous clustering. A wellknown physical clustering model, the Cahn Hilliard model which describes phase separation, is modified to reflect the properties of the data to be visualized. Clusters are defined implicitly as connected components of the positivity set of a density function. An evolution equation for this function is obtained as a suitable gradient flow of an underlying anisotropic energy functional. Here, time serves as the scale parameter. The evolution is characterized by a successive coarsening of patterns — the actual clustering — during which the underlying simulation data specifies preferable pattern boundaries. We introduce specific physical quantities in the simulation to control the shape, orientation and distribution of the clusters, as a function of the underlying flow field. In addition the model is expanded involving elastic effects. Thereby in early stages of the evolution shear layer type representation of the flow field can be generated, whereas for later stages the distribution of clusters can be influenced. Furthermore, we incorporate upwind ideas to give the clusters an oriented drop–shaped appearance. Here we discuss the applicability of this new type of approach mainly for flow fields, where the cluster energy penalizes cross streamline boundaries. However, the method also carries provisions for other fields as well. The clusters can be displayed directly as a flow texture. Alternatively, the clusters can be visualized by iconic representations, which are positioned by using a skeletonization algorithm. 1
On Mathematical Models For Phase Separation In Elastically Stressed Solids
, 2000
"... Contents 1. Introduction 2 2. The diffuse interface model 7 3. Existence for the diffuse interface system 12 3.1. The gradient flow structure 12 3.2. Assumptions 15 3.3. Weak solutions 16 3.4. The implicit time discretisation 17 3.5. Uniform estimates 21 3.6. Proof of the existence theorem 25 3.7. ..."
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Cited by 14 (6 self)
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Contents 1. Introduction 2 2. The diffuse interface model 7 3. Existence for the diffuse interface system 12 3.1. The gradient flow structure 12 3.2. Assumptions 15 3.3. Weak solutions 16 3.4. The implicit time discretisation 17 3.5. Uniform estimates 21 3.6. Proof of the existence theorem 25 3.7. Uniqueness for homogeneous linear elasticity 26 4. Logarithmic free energy 29 4.1. A regularised problem 32 4.2. Higher integrability for the strain tensor 36 4.3. Higher integrability for the logarithmic free energy 42 4.4. Proof of the existence theorem 45 5. The sharp interface limit 46 5.1. The \Gammalimit of the elastic GinzburgLandau energies 52 5.2. EulerLagrange equation for the sharp interface functional 60 6. The GibbsThomson equation as a singular limit in the scalar case 70 7. Discussion 79 8. Appendix 81 9. Notation 86 References 90 1 1. Introduction We study a mathematical model describing phase separation in multi component alloy
The CahnHilliard Equation with Elasticity  Finite Element Approximation and Qualitative Studies
 J. Interphases and Free Boundaries
"... We consider the CahnHilliard equation  a fourthorder, nonlinear parabolic diffusion equation describing phase separation of a binary alloy which is quenched below a critical temperature. The occurrence of two phases is due to a nonconvex double well free energy. The evolution initially leads ..."
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Cited by 10 (7 self)
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We consider the CahnHilliard equation  a fourthorder, nonlinear parabolic diffusion equation describing phase separation of a binary alloy which is quenched below a critical temperature. The occurrence of two phases is due to a nonconvex double well free energy. The evolution initially leads to a very fine microstructure of regions with different phases which tend to become coarser at later times. The resulting phases might have different elastic properties caused by a different lattice spacing. This effect is not reflected by the standard CahnHilliard model. Here, we discuss an approach which contains anisotropic elastic stresses by coupling the expanded diffusion equation with a corresponding quasistationary linear elasticity problem for the displacements on the microstructure. Convergence and a discrete energy decay property are stated for a finite element discretization. An appropriate timestep scheme based on the strongly Astable scheme and a spatial grid adaptation by refining and coarsening improve the algorithms efficiency significantly. Various numerical simulations outline different qualitative effects of the generalized model. Finally, a surprising stabilizing effect of the anisotropic elasticity is observed in the limit case of a vanishing fourth order term, originally representing interfacial energy. 1.
The viscous CahnHilliard equation: Morse decomposition and structure of the global attractor
 Trans Amer Math Soc
, 1999
"... In this paper a partial Morse decomposition of the stationary solutions of the onedimensional viscous Cahn{Hilliard equation is established by explicit energy calculations. Strong nondegeneracy of the stationary solutions is proven away from turning points and points of bifurcation from the homogen ..."
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Cited by 10 (0 self)
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In this paper a partial Morse decomposition of the stationary solutions of the onedimensional viscous Cahn{Hilliard equation is established by explicit energy calculations. Strong nondegeneracy of the stationary solutions is proven away from turning points and points of bifurcation from the homogeneous state and the dimension of the unstable manifold is calculated for all stationary states. In the unstable case, the ow on the global attractor is shown to be semiconjugate to the ow on the global attractor of the Cha eeInfante equation, and in the metastable case close to the nonlocal reaction{di usion limit, a partial description of the structure of the global attractor is obtained by connection matrix arguments, employing a partial energy ordering and the existence of a weak lap number principle. 1
On CahnHilliard Systems with Elasticity
 Proc. Roy. Soc. Edinburgh, 133 A
"... Elastic eects can have a pronounced eect on the phase separation process in solids. The classical GinzburgLandau energy can be modi ed to account for such elastic interactions. The evolution of the system is then governed by diusion equations for the concentrations of the alloy components and by a ..."
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Cited by 8 (1 self)
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Elastic eects can have a pronounced eect on the phase separation process in solids. The classical GinzburgLandau energy can be modi ed to account for such elastic interactions. The evolution of the system is then governed by diusion equations for the concentrations of the alloy components and by a quasistatic equilibrium for the mechanical part. The resulting system of equations is ellipticparabolic and can be understood as a generalisation of the CahnHilliard equation. In this paper we give a derivation of the system and prove an existence and uniqueness result for it. 1.
An impact of stochastic dynamic boundary conditions on the evolution of the CahnHilliard system
, 2006
"... ..."
Efficient preconditioners for large scale binary CahnHilliard models
 COMPUTATIONAL METHODS IN APPLIED MATHEMATICS
"... In this work we consider preconditioned iterative solution methods for numerical simulations of multiphase flow problems, modelled by the CahnHilliard equation. We focus on diphasic flows and the construction and efficiency of a preconditioner for the algebraic systems arising from finite element d ..."
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Cited by 5 (1 self)
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In this work we consider preconditioned iterative solution methods for numerical simulations of multiphase flow problems, modelled by the CahnHilliard equation. We focus on diphasic flows and the construction and efficiency of a preconditioner for the algebraic systems arising from finite element discretizations in space and the thetamethod in time. The preconditioner utilizes to a full extent the algebraic structure of the underlying matrices and exhibits optimal convergence and computational complexity properties. Large scale umerical experiments are included as well as performance comparisons with other solution methods.
A Continuous Clustering Method for Vector Fields
, 2000
"... A new method for the simplification of flow fields is presented. It is based on continuous clustering. A wellknown physical clustering model, the Cahn Hillard model which describes phase separation, is modified to reflect the properties of the data to be visualized. Clusters are defined implicitly ..."
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Cited by 5 (0 self)
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A new method for the simplification of flow fields is presented. It is based on continuous clustering. A wellknown physical clustering model, the Cahn Hillard model which describes phase separation, is modified to reflect the properties of the data to be visualized. Clusters are defined implicitly as connected components of the positivity set of a density function. An evolution equation for this function is obtained as a suitable gradient flow of an underlying anisotropic energy functional. Here, time serves as the scale parameter. The evolution is characterized by a successive coarsening of patterns  the actual clustering  and meanwhile the underlying simulation data specifies preferable pattern boundaries. Here we discuss the applicability of this new type of approach mainly for flow fields, where the cluster energy penalizes cross streamline boundaries, but the method also carries provisions in other fields as well. The clusters are visualized via iconic representations. A skeletonization algorithm is used to find suitable positions for the icons.
Numerical analysis of the CahnHilliard equation and approximation for the HeleShaw problem, Part II: error analysis and convergence of the interface
 SIAM J. Numer. Anal
"... Abstract. In this first part of a series, we propose and analyze, under minimum regularity assumptions, a semidiscrete (in time) scheme and a fully discrete mixed finite element scheme for the CahnHilliard equation ut + ∆(ε∆u − ε−1f(u)) = 0 arising from phase transition in materials science, wher ..."
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Cited by 4 (2 self)
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Abstract. In this first part of a series, we propose and analyze, under minimum regularity assumptions, a semidiscrete (in time) scheme and a fully discrete mixed finite element scheme for the CahnHilliard equation ut + ∆(ε∆u − ε−1f(u)) = 0 arising from phase transition in materials science, where ε is a small parameter known as an “interaction length”. The primary goal of this paper is to establish some useful a priori error estimates for the proposed numerical methods, in particular, by focusing on the dependence of the error bounds on ε. Quasioptimal order error bounds are shown for the semidiscrete and fully discrete schemes under different constraints on the mesh size h and the local time step size km of the stretched time grid, and minimum regularity assumptions on the initial function u0 and domain Ω. In particular, all our error bounds depend on 1 only in some lower polynomial order for small ε. The cruxes of the analysis are to establish ε stability estimates for the discrete solutions, to use a spectrum estimate result of Alikakos and Fusco [3] and Chen [15], and to establish a discrete counterpart of it for a linearized CahnHilliard operator to handle the nonlinear term on the stretched time grid. It is this polynomial dependency of the