This work is concerned with the proof of \emph{a posteriori} error estimates for fully-discrete Galerkin approximations of the Allen-Cahn equation in two and three spatial dimensions. The numerical method comprises of the backward Euler method combined with conforming finite elements in space. For this method, we prove conditional type \emph{a posteriori} error estimates in the $L^{}_4(0,T;L^{}_4(\Omega))$-norm that depend polynomially upon the inverse of the interface length $\epsilon$. The derivation relies crucially on the availability of a spectral estimate for the linearized Allen-Cahn operator about the approximating solution in conjunction with a continuation argument and a variant of the elliptic reconstruction. The new analysis also appears to improve variants of known \emph{a posteriori} error bounds in $L_2(H^1)$, $L_\infty^{}(L_2^{})$-norms in certain regimes.

中文翻译：

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Allen--Cahn 问题的后验误差估计

这项工作涉及证明在两个和三个空间维度中 Allen-Cahn 方程的完全离散 Galerkin 近似的 \emph {a后验} 误差估计。数值方法包括后向欧拉法与空间中的一致有限元相结合。对于这种方法，我们证明了 $L^{}_4(0,T;L^{}_4(\Omega))$-norm 中的条件类型 \emph{a后验} 误差估计，其多项式依赖于接口长度 $\epsilon$。推导主要依赖于线性化 Allen-Cahn 算子关于近似解的谱估计的可用性以及连续参数和椭圆重建的变体。新的分析似乎也改进了 $L_2(H^1)$ 中已知的 \emph{a后验} 误差范围的变体，