Fractional vector calculus is the building block of the fractional partial differential equations that model non-local or long-range phenomena, e.g., anomalous diffusion, fractional electromagnetism, and fractional advection-dispersion. In this work, we reformulate a type of fractional vector calculus that uses Caputo fractional partial derivatives and discretize this reformulation using discrete exterior calculus on a cubical complex in the structure-preserving way, meaning that the continuous-level properties $\operatorname{curl}^\alpha \operatorname{grad}^\alpha = \mathbf{0}$ and $\operatorname{div}^\alpha \operatorname{curl}^\alpha = 0$ hold exactly on the discrete level. We discuss important properties of our fractional discrete exterior derivatives and verify their second-order convergence in the root mean square error numerically. Our proposed discretization has the potential to provide accurate and stable numerical solutions to fractional partial differential equations and exactly preserve fundamental physics laws on the discrete level regardless of the mesh size.

翻译：分向量矢量计算是分片部分方程式的构件,这些方程式模拟非局部或长程现象,例如异常扩散、分子电磁学和分片反演分布。在这项工作中,我们重新配置了一种使用Caputo分片衍生物的分数矢量计算器,并将这种重新配制使用离散外部微量计算器在结构-保留方式的立方体复合体上进行分解,这意味着连续级属性 $\operatorname{curl ⁇ alpha\operatorname{grad{ ⁇ alpha=\mathbf{0}$和$\operatorname{div ⁇ pha\operpha_curalpha_crogatorname{curl ⁇ alpha=0$=0$完全维持在离散水平上。我们讨论我们分数离离式外部衍生物分解衍生物的重要特性,并用数字方式核查它们第二序的趋同。我们提议的离式分数法有可能为分片部分差异部分差异方方方方方方方方程式提供准确和稳定的数字解决方案。