On a Class of Sixth-order Cahn-Hilliard Type Equations with Logarithmic Potential

We consider a class of six-order Cahn-Hilliard equations with logarithmic type potential. This system is closely connected with some important phase-field models relevant in different applications, for instance, the functionalized Cahn-Hilliard equation that describes phase separation in mixtures of amphiphilic molecules in solvent, and the Willmore regularization of Cahn-Hilliard equation for anisotropic crystal and epitaxial growth. The singularity of the configuration potential guarantees that the solution always stays in the physical relevant domain [-1,1]. Meanwhile, the resulting system is characterized by some highly singular diffusion terms that make the mathematical analysis more involved. We prove existence and uniqueness of global weak solutions and show their parabolic regularization property for any positive time. Besides, we investigate long-time behavior of the system, proving existence of the global attractor for the associated dynamical process in a suitable complete metric space.Comment: 35 page

Local Discontinuous Galerkin Methods for the Functionalized Cahn–Hilliard Equation

© 2014, Springer Science+Business Media New York. In this paper, we develop a local discontinuous Galerkin (LDG) method for the sixth order nonlinear functionalized Cahn–Hilliard (FCH) equation. We address the accuracy and stability issues from simulating high order stiff equations in phase-field modeling. Within the LDG framework, various boundary conditions associated with the background physics can be naturally implemented. We prove the energy stability of the LDG method for the general nonlinear case. A semi-implicit time marching method is applied to remove the severe time step restriction (Δt∼O(Δx6)) for explicit methods. The h-p adaptive capability of the LDG method allows for capturing the interfacial layers and the complicated geometric structures of the solution with high resolution. To enhance the efficiency of the proposed approach, the multigrid (MG) method is used to solve the system of linear equations resulting from the semi-implicit temporal integration at each time step. We show numerically that the MG solver has mesh-independent convergence rates. Numerical simulation results for the FCH equation in two and three dimensions are provided to illustrate that the combination of the LDG method for spatial approximation, semi-implicit temporal integration with the MG solver provides a practical and efficient approach when solving this family of problems

Linearly Preconditioned Nonlinear Solvers for Phase Field Equations Involving p-Laplacian Terms

Phase field models are usually constructed to model certain interfacial dynamics. Numerical simulations of phase-field models require long time accuracy, stability and therefore it is necessary to develop efficient and highly accurate numerical methods. In particular, the unconditionally energy stable , unconditionally solvable, and accurate schemes and fast solvers are desirable. In this thesis, We describe and analyze preconditioned steepest descent (PSD) solvers for fourth and sixth-order nonlinear elliptic equations that include p-Laplacian terms on periodic domains in 2 and 3 dimensions. Such nonlinear elliptic equations often arise from time discretization of parabolic equations that model various biological and physical phenomena, in particular, liquid crystals, thin film epitaxial growth and phase transformations. The analyses of the schemes involve the characterization of the strictly convex energies associated with the equations. We first give a general framework for PSD in Hilbert spaces. Based on certain reasonable assumptions of the linear pre-conditioner, a geometric convergence rate is shown for the nonlinear PSD iteration. We then apply the general theory to the fourth and sixth-order problems of interest, making use of Sobolev embedding and regularity results to confirm the appropriateness of our pre-conditioners for the regularized p-Lapacian problems. The results include a sharper theoretical convergence result for p-Laplacian systems compared to what may be found in existing works. We demonstrate rigorously how to apply the theory in the finite dimensional setting using finite difference discretization methods. Based on the PSD framework, we also proposed two efficient and practical Preconditioned Nonlinear Conjugate Gradient (PNCG) solvers. The main idea of the preconditioned solvers is to use a linearized version of the nonlinear operator as a metric for choosing the initial search direction. And the hybrid conjugate directions as the following search direction. In order to make the proposed solvers and scheme much more practical, we also investigate an adaptive time stepping strategy for time dependent problems. Numerical simulations for some important physical application problems – including thin film epitaxy with slope selection, the square phase field crystal model and functionalized Cahn-Hilliard equation – are carried out to verify the efficiency of the schemes and solvers