A finite element scheme for an entirely fractional Allen-Cahn equation with non-smooth initial data is introduced and analyzed. In the proposed nonlocal model, the Caputo fractional in-time derivative and the fractional Laplacian replace the standard local operators. Piecewise linear finite elements and convolution quadratures are the basic tools involved in the presented numerical method. Error analysis and implementation issues are addressed together with the needed results of regularity for the continuous model. Also, the asymptotic behavior of solutions, for a vanishing fractional parameter and usual derivative in time, is discussed within the framework of the Gamma-convergence theory.

中文翻译：

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完全分数阶 Allen-Cahn 方程的数值近似

介绍并分析了具有非光滑初始数据的全分数Allen-Cahn方程的有限元方案。在提议的非局部模型中，Caputo 分数阶时间导数和分数阶拉普拉斯算子取代了标准的局部算子。分段线性有限元和卷积正交是所提出的数值方法中涉及的基本工具。错误分析和实现问题与连续模型所需的规律性结果一起解决。此外，在 Gamma 收敛理论的框架内讨论了解的渐近行为，对于消失的分数参数和通常的时间导数。