1 research outputs found
Extending discrete exterior calculus to a fractional derivative
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 -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.Comment: 18 pages, 11 figures. Work is to be presented at Solid and Physical
Modeling 201