[Paper Review] A posteriori error estimates for finite element approximations of the Cahn-Hilliard equation and the Hele-Shaw flow

Xiaobing Feng, Haijun Wu|ArXiv.org|Aug 15, 2007

Solidification and crystal growth phenomena40 references21 citations

TL;DR

This paper develops a posteriori error estimates of residual type for finite element approximations of the Cahn-Hilliard equation and its sharp interface limit, the Hele-Shaw flow. The key contribution is a robust error bound that depends on $\varepsilon^{-1}$ only in low polynomial order, enabling an effective adaptive finite element algorithm that efficiently resolves the diffuse interface with optimal convergence rates.

ABSTRACT

This paper develops a posteriori error estimates of residual type for conforming and mixed finite element approximations of the fourth order Cahn-Hilliard equation $u_t+\De\bigl(\eps \De u-\eps^{-1} f(u)\bigr)=0$. It is shown that the {\it a posteriori} error bounds depends on $\eps^{-1}$ only in some low polynomial order, instead of exponential order. Using these a posteriori error estimates, we construct an adaptive algorithm for computing the solution of the Cahn-Hilliard equation and its sharp interface limit, the Hele-Shaw flow. Numerical experiments are presented to show the robustness and effectiveness of the new error estimators and the proposed adaptive algorithm.

Motivation & Objective

• To develop reliable a posteriori error estimators for conforming and mixed finite element methods applied to the Cahn-Hilliard equation.
• To establish error bounds that depend on $\varepsilon^{-1}$ only in polynomial order, avoiding exponential dependence.
• To construct an adaptive finite element algorithm based on the a posteriori estimates for both the Cahn-Hilliard equation and its Hele-Shaw limit.
• To demonstrate the robustness and effectiveness of the error estimators and adaptive algorithm through numerical experiments.

Proposed method

• Derive residual-type a posteriori error estimates for conforming and mixed finite element approximations of the Cahn-Hilliard equation.
• Analyze the dependence of error bounds on the small parameter $\varepsilon$, showing polynomial rather than exponential scaling in $\varepsilon^{-1}$.
• Construct an adaptive algorithm that refines elements based on the a posteriori error indicators to resolve the diffuse interface accurately.
• Use the chemical potential $w^\varepsilon = -\varepsilon \Delta u^\varepsilon + \varepsilon^{-1}f(u^\varepsilon)$ as a key quantity in the error estimation and adaptation process.
• Apply the method to both the Cahn-Hilliard equation and its sharp interface limit (Hele-Shaw flow) via asymptotic analysis as $\varepsilon \to 0$.
• Implement and validate the adaptive algorithm on benchmark problems with varying initial conditions and $\varepsilon$ values.

Experimental results

Research questions

• RQ1 Can a posteriori error estimators be constructed for finite element solutions of the Cahn-Hilliard equation that remain robust as $\varepsilon \to 0$?
• RQ2 Does the error bound depend on $\varepsilon^{-1}$ in exponential or only polynomial order?
• RQ3 Can an adaptive finite element method be designed to efficiently resolve the thin diffuse interface in the Cahn-Hilliard model?
• RQ4 What convergence rate is achieved by the adaptive method for the zero level set of the solution?
• RQ5 How does the adaptive method compare to uniform refinement in terms of degrees of freedom and accuracy?

Key findings

• The a posteriori error estimators exhibit only polynomial dependence on $\varepsilon^{-1}$, avoiding exponential growth and ensuring robustness for small $\varepsilon$.
• The adaptive algorithm successfully resolves the diffuse interface with significantly reduced degrees of freedom compared to uniform refinement.
• For Test 1, the convergence rate of the zero level set is approximately $O(1/\mathcal{N}^2)$, where $\mathcal{N}$ is the number of degrees of freedom.
• With $TOL=0.01$, the adaptive method achieves $\mathcal{N}_{0.01} = 12,565$ DOFs at $t=0.01$, while uniform refinement would require about 1.18 million DOFs.
• In Test 2, the adaptive method uses only 2,520 initial elements with minimum area $1.2207 \times 10^{-4}$, whereas uniform refinement would require ~32,768 elements.
• In Test 3, the adaptive method uses 4,072 initial elements with minimum area $3.0518 \times 10^{-5}$, while uniform refinement would require ~131,072 elements.