Robust a posteriori error control for the Allen-Cahn equation with variable mobility

Abstract

In this work, we derive a $γ$-robust a posteriori error estimator for finite element approximations of the Allen-Cahn equation with variable non-degenerate mobility. The estimator utilizes spectral estimates for the linearized steady part of the differential operator as well as a conditional stability estimate based on a weighted sum of Bregman distances, based on the energy and a functional related to the mobility. A suitable reconstruction of the numerical solution in the stability estimate leads to a fully computable estimator.

Figures (6)

• Figure 1: Time evolution of $\phi$ at different time steps. The topological singularity, here the collapse of the annulus to a sphere, happens around $t\approx 0.072$.
• Figure 2: Experimental scaling of the residuals with respect to $\gamma$. Discretisation parameters $\gamma_k=2^{-k},h_{\min}\approx 0.03,h_{\max} \approx 0.065,\tau\approx 5\cdot 10^{-4},$ for $k=2,\ldots,11$.
• Figure 3: Experimental scaling of the residuals with respect to $h$. Discretisation parameters $\gamma=2^{-4},h_{k,\min}\approx 2^{-k},h_{k,\max}\approx 1.5\cdot 2^{-k} ,\tau=2.5\cdot 10^{-4}$ for $k=0,\ldots,4$.
• Figure 4: Experimental scaling of the residuals with respect to $h$. Discretisation parameters $\gamma=2^{-4},h_{\min}\approx 0.025,h_{\max} \approx 0.058,\tau_k=2^{-k}$ for $k=4,\ldots,11$.
• Figure 5: Experimental scaling of the integrated eigenvalue $\int_0^t \lambda_+$ with respect to $\gamma$. Discretisation parameters $\gamma_k=2^{-k},h_{\min}\approx 0.066,h_{\max} \approx 0.135,\tau=3\cdot 10^{-4}$ for $k=2,\ldots,12$.

Theorems & Definitions (15)

• Definition 1
• Lemma 2
• proof
• Lemma 3
• proof
• Remark 4
• Lemma 5
• proof
• Lemma 6
• proof