Structure-Preserving Discretization of Fractional Vector Calculus using Discrete Exterior Calculus

TL;DR

This paper introduces a structure-preserving discretization method for fractional vector calculus using discrete exterior calculus, ensuring key properties hold exactly on the discrete level and enabling accurate solutions to fractional PDEs.

Contribution

It reformulates fractional vector calculus with Caputo derivatives and discretizes it via discrete exterior calculus, preserving fundamental properties exactly at the discrete level.

Findings

Second-order convergence verified numerically

Exact preservation of continuous properties on discrete level

Potential for accurate, stable numerical solutions to fractional PDEs

Abstract

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.