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

Energy Stable Numerical Schemes for Ternary Cahn-Hilliard System

We present and analyze a uniquely solvable and unconditionally energy stable numerical scheme for the ternary Cahn-Hilliard system, with a polynomial pattern nonlinear free energy expansion. One key difficulty is associated with presence of the three mass components, though a total mass constraint reduces this to two components. Another numerical challenge is to ensure the energy stability for the nonlinear energy functional in the mixed product form, which turns out to be non-convex, non-concave in the three-phase space. to overcome this subtle difficulty, we add a few auxiliary terms to make the combined energy functional convex in the three-phase space, and this, in turn, yields a convex-concave decomposition of the physical energy in the ternary system. Consequently, both the unique solvability and the unconditional energy stability of the proposed numerical scheme are established at a theoretical level. in addition, an optimal rate convergence analysis in the ℓ∞(0,T;HN-1)∩ℓ2(0,T;HN1) norm is provided, with Fourier pseudo-spectral discretization in space, which is the first such result in this field. to deal with the nonlinear implicit equations at each time step, we apply an efficient preconditioned steepest descent (PSD) algorithm. a second order accurate, modified BDF scheme is also discussed. a few numerical results are presented, which confirm the stability and accuracy of the proposed numerical scheme