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

Xiaobing Feng, Steven Wise

arXiv:1105.1148v2math.NAcs.NA

TL;DR

Moving interfaces in Hele-Shaw flow can undergo topological changes that complicate classical solution and computation. The paper analyzes the Darcy-Cahn-Hilliard diffuse-interface model and develops a fully discrete finite element method. It establishes stability, solvability, and convergence results, including global-in-time weak-solution existence in two and three dimensions.

  • Problem

    Topological changes in moving Hele-Shaw interfaces can end classical solutions, making generalized solutions delicate to define and compute.

    Source: §1 · p. 2

  • Method

    The paper analyzes the Darcy-Cahn-Hilliard model and proposes a fully discrete finite element method for its approximation.

    Source: §1 · p. 4

  • Results

    The method is unconditionally energy stable and uniquely solvable at each time step, and its finite element solutions converge to weak solutions.

    Source: §1 · p. 4, §4 · p. 22

  • Takeaways & Limitations

    The convergence result provides a constructive proof of global-in-time weak-solution existence in two and three dimensions.

    Source: §1 · p. 4

    The reported error bound is not useful for small ε; deriving a bound with only low-polynomial-order dependence on ε^-1 remains ongoing work.

    Source: §4 · p. 22

Abstract

from arXiv · show

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.

1. Introduction.

The paper develops a diffuse-interface approach to Hele-Shaw flow, where moving interfaces can undergo topological changes that challenge classical solutions. It analyzes the Darcy-Cahn-Hilliard model and its finite element approximation.

1. Introduction.

  • Hele-Shaw flow models viscous fluids between parallel plates, and its governing equations are useful for studying interfacial dynamics.The paper notes that the equations’ relative simplicity makes these flows test cases for mathematical theory and numerical methods.Source: §1 · pp. 1–2
  • Topological changes can end classical solutions of the moving-interface problem, making generalized solutions delicate to define and compute.The diffuse-interface approach represents the interface as a finite-thickness layer whose phase field varies smoothly.Source: §1 · p. 2
  • The DCH system is a special case of the BHSCH model; formal asymptotics showed BHSCH solutions converge to Hele-Shaw solutions as the interfacial parameter tends to zero.The paper notes that its pressure uses a different scaling from the BHSCH model.Source: §1 · p. 3
  • The fully discrete method conserves mass and satisfies a discrete energy law that mimics the model’s energy law uniformly in ε.The paper uses the discrete energy law to prove unconditional energy stability and unique solvability at every time step.Source: §1 · pp. 3–4
  • The authors use a nonlinear multigrid iterative method for the finite element equations and report numerical experiments validating theory and demonstrating the method’s effectiveness.The method also yields a constructive proof of global-in-time weak-solution existence in two and three dimensions.Source: §1 · p. 4

2. PDE analysis.

The PDE analysis formulates weak solutions for the Darcy-Cahn-Hilliard system and studies their energy behavior, regularity, and uniqueness. It also relates these properties to the variables and function spaces used in the analysis.

2. PDE analysis.

  • The analysis approximates pressure and phase variables directly while computing velocity as an auxiliary variable when needed.It uses a standard finite element method for pressure and a mixed finite element method for the phase equation.Source: §2 · p. 4
  • The DCH system’s energy law is the basis for the weak-solution formulation and is used to establish energy behavior.The proof uses Steklov averages to handle a time derivative that cannot directly serve as a test function.Source: §2 · pp. 4–7
  • The regularity of ϕ ensures ϕ ∈ C0([0,T];L2(Ω)), so the initial condition ϕ(0)=ϕε0 is meaningful.The paper attributes this continuity to the Aubin-Lions lemma.Source: §2 · p. 5
  • The solution regularities are chosen to match those suggested by the energy law, and the resulting product space is the DCH system’s energy space.The paper notes these are not the minimum regularities needed to make every term well-defined.Source: §2 · p. 5
  • Weak solutions are unique in the function class F, which imposes additional integrability and time-regularity conditions.The class requires ∇p + γϕ∇µ ∈ L^(12/(6−d))((0,T);L2(Ω)), µ ∈ L^(12/(6−d))((0,T);H1(Ω)), and ϕt ∈ L2((0,T);(H1(Ω))∗).Source: §2 · p. 8

3. Fully discrete finite element method.

The section formulates a fully discrete finite element scheme and establishes its stability and solvability. Its analysis uses energy estimates and convexity, rather than an inf-sup condition, to prove stability of the mixed finite element triple.

3.1. Formulation of the finite element method.

  • The mixed finite element triple (Wh, Vh, Vh) is shown to be stable using energy estimates and convexity rather than an inf-sup condition.Source: §3.1 · p. 13

3.2. Well-posedness of the finite element method.

  • The scheme is uniquely solvable for any mesh parameters τ and h and any phase parameter ε.The proof reformulates each time step as minimization of a strictly convex, coercive functional.Source: §3.1 · p. 13, §3.2 · pp. 14–15
  • The scheme conserves mass and satisfies a discrete energy law that yields uniform-in-ε, h, and τ a priori estimates.The mass conservation property matches that of the phase field model.Source: §3.2 · pp. 16–17

4. Convergence analysis.

The convergence analysis shows that, under bounded initial discrete energy, the fully discrete solutions have an accumulation point that is a weak solution. It also establishes weak-solution existence and uniqueness in the stated function class, with a domain restriction that can be relaxed to bounded Lipschitz domains.

4. Convergence analysis.

  • For fixed ε > 0 on polygonal or polyhedral domains in d = 2, 3, bounded initial discrete energy yields an accumulation point that is a weak solution.The initial-energy bound is independent of h.Source: §4 · p. 20
  • If weak solutions are unique, the whole finite element sequence converges to the unique weak solution.The proof establishes that every convergent subsequence has a weak-solution limit.Source: §4 · p. 22
  • The polygonal or polyhedral domain assumption is technical; the convergence conclusions also hold on bounded Lipschitz domains with modified finite element methods.The paper cites numerical integration or boundary approximation as ways to construct those methods.Source: §4 · p. 22
  • The convergence result provides a constructive existence proof, and weak solutions are unique in the function class F.The existence and uniqueness conclusion follows from Theorem 4.2 and Theorem 2.4.Source: §4 · pp. 17, 22
  • The scheme converges optimally in the energy norm, but its error bound is not useful for small ε; a polynomial-order bound remains ongoing work.The paper says a more useful bound would depend on ε^-1 in a low polynomial order.Source: §4 · p. 22

5. Numerical experiments.

The experiments test numerical convergence and simulate spinodal decomposition. They report optimal convergence rates, energy decrease, and very small mass variation in the tested simulations.

Convergence with source terms

  • With artificial source terms, the L2 error test confirms optimal second-order convergence and the H1 error test confirms optimal first-order convergence.The L2 and H1 tests use quadratic and linear refinement paths, respectively.Source: §5 · p. 23

Cauchy convergence

  • The L2 Cauchy analysis confirms second-order convergence, while the H1 Cauchy analysis confirms first-order convergence.The Cauchy differences compare solutions at successive spatial and temporal resolutions.Source: §5 · p. 24

Spinodal decomposition

  • For the spinodal-decomposition simulations, at least at early times, energy decreases faster and the coarsening process appears accelerated as excess surface tension γ increases; energy decreases at each time step.Source: §5 · p. 25, Figure 5.2 · p. 26

Mass conservation

  • For γ = 0.01, mass variation over the simulation is 1 × 10^-12, close to the multigrid stopping tolerance of 1 × 10^-12.The initial average of ϕ is approximately −0.1; theoretically, mass is expected to remain unchanged between time steps.Source: §5 · pp. 25–26
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