Analysis of a Darcy-Cahn-Hilliard Diffuse Interface Model for the Hele-Shaw Flow and its Fully Discrete Finite Element Approximation

In this paper we present PDE and finite element analyses for a system of partial differential equations (PDEs) consisting of the Darcy equation and the Cahn-Hilliard equation, which arises as a diffuse interface model for the two phase Hele-Shaw flow. We propose a fully discrete implicit finite element method for approximating the PDE system, which consists of the implicit Euler method combined with a convex splitting energy strategy for the temporal discretization, the standard finite element discretization for the pressure and a split (or mixed) finite element discretization for the fourth order Cahn-Hilliard equation. It is shown that the proposed numerical method satisfies a mass conservation law in addition to a discrete energy law that mimics the basic energy law for the Darcy-Cahn-Hilliard phase field model and holds uniformly in the phase field parameter $\epsilon$. With help of the discrete energy law, we first prove that the fully discrete finite method is unconditionally energy stable and uniquely solvable at each time step. We then show that, using the compactness method, the finite element solution has an accumulation point that is a weak solution of the PDE system. As a result, the convergence result also provides a constructive proof of the existence of global-in-time weak solutions to the Darcy-Cahn-Hilliard phase field model in both two and three dimensions. Finally, we propose a nonlinear multigrid iterative algorithm to solve the finite element equations at each time step. Numerical experiments based on the overall solution method of combining the proposed finite element discretization and the nonlinear multigrid solver are presented to validate the theoretical results and to show the effectiveness of the proposed fully discrete finite element method for approximating the Darcy-Cahn-Hilliard phase field model.Comment: 30 pages, 4 tables, 2 figure

Positivity-preserving, energy stable numerical schemes for the Cahn-Hilliard equation with logarithmic potential

We present and analyze finite difference numerical schemes for the Allen Cahn/Cahn-Hilliard equation with a logarithmic Flory Huggins energy potential. Both the first order and second order accurate temporal algorithms are considered. In the first order scheme, we treat the nonlinear logarithmic terms and the surface diffusion term implicitly, and update the linear expansive term and the mobility explicitly. We provide a theoretical justification that, this numerical algorithm has a unique solution such that the positivity is always preserved for the logarithmic arguments. In particular, our analysis reveals a subtle fact: the singular nature of the logarithmic term around the values of $-1$ and 1 prevents the numerical solution reaching these singular values, so that the numerical scheme is always well-defined as long as the numerical solution stays similarly bounded at the previous time step. Furthermore, an unconditional energy stability of the numerical scheme is derived, without any restriction for the time step size. The unique solvability and the positivity-preserving property for the second order scheme are proved using similar ideas, in which the singular nature of the logarithmic term plays an essential role. For both the first and second order accurate schemes, we are able to derive an optimal rate convergence analysis, which gives the full order error estimate. The case with a non-constant mobility is analyzed as well. We also describe a practical and efficient multigrid solver for the proposed numerical schemes, and present some numerical results, which demonstrate the robustness of the numerical schemes

Convergence Analysis and Error Estimates for a Second Order Accurate Finite Element Method for the Cahn-Hilliard-Navier-Stokes System

In this paper, we present a novel second order in time mixed finite element scheme for the Cahn-Hilliard-Navier-Stokes equations with matched densities. The scheme combines a standard second order Crank-Nicholson method for the Navier-Stokes equations and a modification to the Crank-Nicholson method for the Cahn-Hilliard equation. In particular, a second order Adams-Bashforth extrapolation and a trapezoidal rule are included to help preserve the energy stability natural to the Cahn-Hilliard equation. We show that our scheme is unconditionally energy stable with respect to a modification of the continuous free energy of the PDE system. Specifically, the discrete phase variable is shown to be bounded in $\ell^\infty \left(0,T;L^\infty\right)$ and the discrete chemical potential bounded in $\ell^\infty \left(0,T;L^2\right)$, for any time and space step sizes, in two and three dimensions, and for any finite final time $T$. We subsequently prove that these variables along with the fluid velocity converge with optimal rates in the appropriate energy norms in both two and three dimensions.Comment: 33 pages. arXiv admin note: text overlap with arXiv:1411.524