TY - JOUR
T1 - Extending Discrete Exterior Calculus to a Fractional Derivative
AU - Crum, Justin
AU - Levine, Joshua A.
AU - Gillette, Andrew
N1 - Funding Information:
This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research , under Award Number(s) DE-SC-0019039 .
Publisher Copyright:
© 2019 Elsevier Ltd
PY - 2019/9
Y1 - 2019/9
N2 - 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.
AB - 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.
KW - Discrete exterior calculus (DEC)
KW - Fractional derivative
KW - Fractional differential equations
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U2 - 10.1016/j.cad.2019.05.018
DO - 10.1016/j.cad.2019.05.018
M3 - Article
AN - SCOPUS:85065894738
VL - 114
SP - 64
EP - 72
JO - CAD Computer Aided Design
JF - CAD Computer Aided Design
SN - 0010-4485
ER -