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Stability and Convergence of a Second Order Mixed Finite Element Method for the Cahn-Hilliard Equation
Amanda E. Diegel, Cheng Wang, Steven M. Wise
TL;DR
The paper addresses the difficulty of constructing accurate, stable, and solvable numerical schemes for the nonlinear fourth-order Cahn–Hilliard equation. It develops and analyzes a second-order mixed finite element scheme using convex splitting, proving unconditional stability and solvability together with optimal energy-norm convergence. The results include unconditional bounds for the discrete phase variable and chemical potential in two and three dimensions.
Problem
The Cahn–Hilliard equation is challenging to discretize because explicit methods have severe time-step restrictions, while fully implicit methods require potentially large nonlinear algebraic solves.
Method
The paper develops a second-order-in-time, fully discrete mixed finite element scheme based on convex splitting, treating one energy variation implicitly and the other explicitly.
Results
The scheme is unconditionally energy stable and uniquely solvable, provides unconditional L∞(0, T; L∞(Ω)) and L∞(0, T; L2(Ω)) bounds for the phase variable and chemical potential, and converges optimally in appropriate energy norms.
Takeaways & Limitations
The analysis supplies a second-order scheme whose stability, solvability, and energy-norm convergence hold independently of time and space step sizes in two and three dimensions.
Takeaways & Limitations
The stability and error analyses fix ε and do not track how their estimates depend on this interface parameter, which may matter as ε tends to zero.
Abstract
from arXiv · showhide
In this paper we devise and analyze an unconditionally stable, second-order-in-time numerical scheme for the Cahn-Hilliard equation in two and three space dimensions. We prove that our two-step scheme is unconditionally energy stable and unconditionally uniquely solvable. Furthermore, we show that the discrete phase variable is bounded in $L^\infty (0,T;L^\infty)$ and the discrete chemical potential is bounded in $L^\infty (0,T;L^2)$, 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 converge with optimal rates in the appropriate energy norms in both two and three dimensions. We include in this work a detailed analysis of the initialization of the two-step scheme.
1 Introduction
The paper develops a second-order, fully discrete mixed finite element scheme for the Cahn–Hilliard equation that targets stability, solvability, and optimal convergence in two and three dimensions. It addresses the time-step restrictions and analytical gaps affecting existing numerical approaches.
- Motivation and problem: The Cahn–Hilliard equation is a challenging fourth-order nonlinear problem for which explicit methods face severe time-step restrictions and fully implicit methods require large nonlinear solves.Second-order-in-time schemes are less commonly studied and generally harder to analyze than first-order methods.
- Motivation and problem: Existing secant-type schemes can be energy stable without being unconditionally uniquely solvable, limiting their reliability for simulations using very large time steps.Related work also includes schemes lacking formal stability or convergence analyses, or lacking energy-norm second-order accuracy proofs.
- Proposed scheme: The paper proposes a second-order-accurate-in-time, fully discrete mixed finite element scheme closely related to a finite difference scheme.The formulation uses convex splitting, treating the convex-energy variation implicitly and the concave-energy variation explicitly.
- Main contributions: The scheme is unconditionally energy stable, unconditionally uniquely solvable, and optimally convergent in the energy norm independently of time and space step sizes.The analysis also establishes unconditional L∞(0, T; L∞(Ω)) stability for the phase field and L∞(0, T; L2(Ω)) stability for the chemical potential.
- Analysis and validation: The paper proves stability and solvability, derives error estimates under suitable regularity assumptions, and reports numerical tests confirming the predicted convergence rates.It also includes analysis of the two-step scheme's initialization through the scheme definition and subsequent results.
2 A Mixed Finite Element Convex Splitting Scheme
The paper defines a second-order mixed finite element convex-splitting scheme with a separately analyzed initialization. Its stability, solvability, and a priori estimates hold without restrictions on time or space step sizes under stated assumptions.
- Scheme definition: The scheme uses a second-order mixed finite element formulation with a separate initialization process for its multistep structure.The initial phase variable uses a Ritz projection, and the scheme also requires initial chemical-potential data.
- Scheme definition: The initialization scheme is uniquely solvable for any mesh parameters h and τ and any model parameters.This establishes unconditional solvability for the first step as well as the main scheme.
- Unconditional stability: The scheme is unconditionally energy stable and uniquely solvable, with stability estimates independent of both h and τ.The first-step energy law and subsequent a priori estimates are stated for arbitrary positive h and τ.
- Assumptions and scope: The error-analysis framework assumes stability conditions on the initial data and fixes the interface parameter ε without tracking its dependence in the estimates.The paper notes that dependence on ε may matter if ε tends to zero.
- A priori estimates: The analysis establishes a priori stability estimates for the discrete solution, including bounds derived using discrete negative norms and Sobolev embeddings in dimensions two and three.The estimates are proved through multiple parts combining initial stability, discrete inequalities, and embeddings.
3 Error Estimates for the Fully Discrete Convex Splitting Scheme
This section develops a rigorous error analysis for the fully discrete convex-splitting scheme under additional regularity assumptions on the weak PDE solution. The analysis combines consistency, finite-element approximation, stability estimates, and a discrete Gronwall argument to obtain convergence estimates.
- The convergence analysis assumes additional regularity of the weak PDE solution and sufficiently regular initial data.
- The proof constructs a key error equation and bounds its consistency, approximation, and stability terms through technical lemmas.
- The main convergence theorem applies when the time step satisfies 0 < τ < τ0 for a sufficiently small τ0.
- The resulting estimates use constants independent of h and τ, although some constants may depend on the final time T through solution regularity.
- A discrete Gronwall inequality converts the intermediate error bounds into the stated convergence result, including an optimal error estimate.
4 Numerical Experiments
The numerical experiments assess the accuracy and reliability of the fully discrete finite element method on a refined square-domain mesh using quadratic finite elements. The reported setup includes a fixed interface parameter and an H1 Cauchy convergence test.
- The experiments use the square domain Ω = (0, 1)^2 with nested regular triangulations formed by subdividing each triangle into four.
- Quadratic P2 finite elements are used for both the phase field and chemical potential, corresponding to q = 2.
- The scheme is solved with ε = 6.25 × 10^-2, and the initialization uses standard nodal interpolation whose error is stated to be optimal.
- Table 1 reports an H1 Cauchy convergence test at final time T = 4.0 × 10^-1 along the specified refinement path.