Mass conservative and energy stable finite difference methods for the quasi-incompressible Navier–Stokes–Cahn–Hilliard system:Primitive variable and projection-type schemes

In this paper we describe two fully mass conservative, energy stable, finite difference methods on a staggered grid for the quasi-incompressible Navier-Stokes-Cahn-Hilliard (q-NSCH) system governing a binary incompressible fluid flow with variable density and viscosity. Both methods, namely the primitive method (finite difference method in the primitive variable formulation) and the projection method (finite difference method in a projection-type formulation), are so designed that the mass of the binary fluid is preserved, and the energy of the system equations is always non-increasing in time at the fully discrete level. We also present an efficient, practical nonlinear multigrid method - comprised of a standard FAS method for the Cahn-Hilliard equation, and a method based on the Vanka-type smoothing strategy for the Navier-Stokes equation - for solving these equations. We test the scheme in the context of Capillary Waves, rising droplets and Rayleigh-Taylor instability. Quantitative comparisons are made with existing analytical solutions or previous numerical results that validate the accuracy of our numerical schemes. Moreover, in all cases, mass of the single component and the binary fluid was conserved up to 10 to -8 and energy decreases in time

Numerical Methods for Deterministic and Stochastic Phase Field Models of Phase Transition and Related Geometric Flows

This dissertation consists of three integral parts with each part focusing on numerical approximations of several partial differential equations (PDEs). The goals of each part are to design, to analyze and to implement continuous or discontinuous Galerkin finite element methods for the underlying PDE problem. Part One studies discontinuous Galerkin (DG) approximations of two phase field models, namely, the Allen-Cahn and Cahn-Hilliard equations, and their related curvature-driven geometric problems, namely, the mean curvature flow and the Hele-Shaw flow. We derive two discrete spectrum estimates, which play an important role in proving the sharper error estimates which only depend on a negative power of the singular perturbation parameter ε [epsilon] instead of an exponential power. It is also proved that the zero level sets of the numerical solutions of the Allen-Cahn equation and the Cahn-Hilliard equation approximate the mean curvature flow and the Hele-Shaw flow respectively. Numerical experiments are carried out to verify the theoretical results and to compare the zero level sets of the numerical solutions and the geometric motions. Part Two focuses on finite element approximations of stochastic geometric PDEs including the phase field formulation of a stochastic mean curvature flow and the level set formulation of the stochastic mean curvature flow. Both formulations give PDEs with gradient-type multiplicative noises. We establish PDE energy laws and the Hölder [Holder] continuity in time for the exact solutions. Moreover, optimal error estimates are derived, and various numerical experiments are carried out to study the interplay of the geometric evolution and gradient-type noises. Part Three studies finite element methods for a quasi-static model of poroelasticity, which is a fluid-solid interaction multiphysics system at pore scale. We reformulate the original multiphysics system into a new system which explicitly reveals the diffusion process and has a built-in mechanism to overcome the locking phenomenon . Fully discrete finite element methods are proposed for approximating the new system. We derive a discrete energy law and optimal error estimates for our finite element methods. Numerical experiments are also provided to verify the theoretical results and to confirm that the locking phenomenon has indeed been overcome

Fast Solvers for Cahn-Hilliard Inpainting

We consider the efficient solution of the modified Cahn-Hilliard equation for binary image inpainting using convexity splitting, which allows an unconditionally gradient stable time-discretization scheme. We look at a double-well as well as a double obstacle potential. For the latter we get a nonlinear system for which we apply a semi-smooth Newton method combined with a Moreau-Yosida regularization technique. At the heart of both methods lies the solution of large and sparse linear systems. We introduce and study block-triangular preconditioners using an efficient and easy to apply Schur complement approximation. Numerical results indicate that our preconditioners work very well for both problems and show that qualitatively better results can be obtained using the double obstacle potential