Extending Discrete Exterior Calculus to a Fractional Derivative. (September 2019)

• Record Type: Journal Article
• Authors: Crum, Justin; Levine, Joshua A.; Gillette, Andrew
• Is Part Of: Computer aided design. Volume 114(2019)
• Page Start: 64
• Page End: 72
• Publication Date: 2019-09
• Subjects: Discrete exterior calculus (DEC) -- Fractional derivative -- Fractional differential equations; Computer-aided design -- Periodicals; Engineering design -- Data processing -- Periodicals; Computer graphics -- Periodicals; Conception technique -- Informatique -- Périodiques; Infographie -- Périodiques; Computer graphics; Engineering design -- Data processing; Periodicals; Electronic journals; 620.00420285
• DOI: 10.1016/j.cad.2019.05.018
• Languages: English
• ISSNs: 0010-4485

Abstract

Abstract: Fractional partial differential equations (FDEs) are used to describe phenomena that involve a "non-local" or "long-range" interaction of some kind. Accurate and practical numerical approximation of their solutions is challenging due to the dense matrices arising from standard discretization procedures. In this paper, we begin to extend the well-established computational toolkit of Discrete Exterior Calculus (DEC) to the fractional setting, focusing on proper discretization of the fractional derivative. We define a Caputo-like fractional discrete derivative, in terms of the standard discrete exterior derivative operator from DEC, weighted by a measure of distance between p -simplices in a simplicial complex. We discuss key theoretical properties of the fractional discrete derivative and compare it to the continuous fractional derivative via a series of numerical experiments. Highlights: Give a new definition of a fractional discrete exterior derivative, based off DEC. Numerically analyze convergence rates on 1-D examples. Numerically investigate results on 2-D examples using.