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Convergence Analysis and Error Estimates for a Second Order Accurate Finite Element Method for the Cahn-Hilliard-Navier-Stokes System
Amanda E. Diegel, Cheng Wang, Xiaoming Wang, Steven M. Wise
TL;DR
The paper addresses the lack of rigorous error analysis for second-order CHNS schemes while retaining energy stability. It introduces a coupled, nearly linear mixed finite element method and proves optimal unconditional convergence estimates, subject to stated regularity and discretization conditions.
Problem
Second-order CHNS schemes can be energy stable and uniquely solvable, but rigorous error analysis and stronger stability estimates remain limited.
Method
The paper combines modified second-order Crank–Nicolson and Adams–Bashforth discretizations in a coupled mixed finite element scheme for matched-density CHNS.
Results
Optimal convergence estimates are obtained in ℓ∞(0,T;H1) for the phase variable and ℓ2(0,T;H1) for the chemical potential, without a τ–h scaling law.
Takeaways & Limitations
The scheme supports second-order CHNS computation with unconditional energy stability and optimal energy-norm convergence in two and three dimensions.
Takeaways & Limitations
The analysis assumes the phase finite-element space is a subspace of the pressure space to retain mass conservation; decoupling these spaces is left for future work.
Abstract
from arXiv · showhide
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.
1 Introduction
The paper develops and analyzes a second-order mixed finite element scheme for the matched-density CHNS system, addressing the difficulty of rigorous error analysis while preserving unconditional energy stability. Its analysis establishes boundedness and optimal, unconditional convergence estimates in two and three dimensions.
- 1 Introduction: The scheme couples modified second-order Crank–Nicolson treatment of Cahn–Hilliard with Crank–Nicolson Navier–Stokes discretization.Second-order Adams–Bashforth extrapolations linearize selected terms while preserving accuracy, energy stability, and solvability.
- 1 Introduction: The method is coupled and almost linear, with only one weak nonlinearity arising from the chemical-potential equation.This distinguishes it from fully decoupled approaches whose convergence analysis is described as more challenging.
- 1 Introduction: A backward-in-time induction estimate and a non-standard discrete Gronwall inequality yield an ℓ∞(0,T;H2) bound for the discrete phase variable.The resulting bound grows at most linearly in time.
- 1 Introduction: The analysis obtains optimal convergence in ℓ∞(0,T;H1) for the phase variable and ℓ2(0,T;H1) for the chemical potential.These estimates are unconditional and require no scaling relation between time step τ and spatial mesh size h.
- 1 Introduction: The scheme is proved unconditionally stable and solvable with respect to both time and space step sizes, followed by rigorous error analysis under suitable PDE regularity assumptions.The paper also reviews discrete Gronwall inequalities used in the analysis.
2 A Second-Order-in-Time, Mixed Finite Element Scheme
The paper defines a second-order mixed finite element scheme and establishes its solvability, mass conservation, unconditional energy stability, and stability estimates for arbitrary time and space step sizes. It then develops discrete estimates for the phase variable using finite-element inequalities, weighted time-step arguments, and discrete Gronwall bounds.
- Scheme definition: The formulation imposes a restrictive relation between the phase-field and pressure spaces; otherwise, mass conservation is lost.An alternate trilinear form can decouple the pressure space from the phase space, but that case is left for future work.
- Scheme definition: The fully discrete second-order convex splitting scheme is uniquely solvable and mass conservative.
- Unconditional energy stability: The scheme is unconditionally energy stable for arbitrary time and space step sizes through a convex decomposition.Its discrete energy law holds for the fully discrete solution and all relevant time levels.
- Unconditional energy stability: Stability estimates hold for any h, τ > 0, with constants independent of h, τ, and T.The analysis defines a modified energy and a discrete Laplacian to obtain these a priori bounds.
- Discrete phase stability: The phase-variable stability analysis combines discrete Gagliardo-Nirenberg inequalities, Sobolev embeddings, weighted time-step estimates, and discrete Gronwall arguments.The resulting estimates are developed for two- and three-dimensional quasi-uniform finite-element meshes under stated initial-stability and regularity assumptions.
3 Error Estimates for the Fully Discrete Scheme
The section derives error estimates for the fully discrete second-order mixed finite element scheme under additional regularity assumptions. Its analysis combines consistency, approximation, stability, and discrete Gronwall arguments to establish convergence with constants independent of discretization parameters, subject to a sufficiently small time step for the main theorem.
- Assumptions: The error analysis assumes additional regularity of the weak solution, including φ ∈ L∞(0,T;W^1,6(Ω)) and higher time regularity.The velocity and pressure also satisfy additional regularity assumptions, and the initial data must support the stated stability estimates.
- Error formulation: The analysis compares the weak solution with the fully discrete scheme at half-integer time steps and derives a coupled error equation for the phase, chemical potential, and velocity.The resulting error equation is defined for 1 ≤ m ≤ M−1 and combines the discrete variational equations with consistency terms.
- Estimate construction: The proof bounds consistency, approximation, diffusion, pressure, and nonlinear convection terms using finite element estimates, inverse inequalities, Sobolev embeddings, and trilinear-form bounds.The convection analysis uses a decomposition in the spirit of Baker’s analysis, with one term vanishing by antisymmetry.
- Stability and convergence: The scheme’s stability estimates hold for arbitrary h and τ, while the main convergence theorem additionally requires 0 < τ < τ0.The theorem’s error constant C(T) is independent of τ and h, although it may depend on T through solution regularity.
- Main result: The final argument combines intermediate lemmas with a discrete Gronwall inequality to obtain optimal error estimates for the second-order splitting scheme.The estimates include a temporal contribution of order τ^4 and a spatial contribution of order h^2q in the displayed error bound.
- Scope of derivation: The error bounds rely on standard finite element spatial approximations and omit some final details involving the triangle inequality.The paper explicitly states that these details are omitted for brevity.
A Some Discrete Gronwall Inequalities
The appendix reviews discrete Gronwall inequalities used in the stability and convergence analysis. It presents both a basic form and a more general form with product factors and parameter restrictions.
- Basic inequality: The appendix introduces a discrete Gronwall inequality for sequences controlled by an explicit accumulated sum.The inequality is stated for fixed final time T and time-step parameters independent of τ and M.
- Application conditions: The appendix emphasizes that the sum on the right-hand side of the basic inequality must remain explicit.This condition motivates the use of the more general inequality in the stability analysis.
- General inequality: A more general inequality is stated with a parameter 0 < α < 1 and product factors of the form dℓ,m.The product bound is used to derive the desired estimate through substitution into the preceding inequality.