A fractional model for anomalous diffusion with increased variability: Analysis, algorithms and applications to interface problems

D'Elia, Marta; Glusa, Christian

doi:10.1002/num.22865

Title: A fractional model for anomalous diffusion with increased variability: Analysis, algorithms and applications to interface problems

Journal Article · Mon Feb 07 00:00:00 EST 2022 · Numerical Methods for Partial Differential Equations

DOI:https://doi.org/10.1002/num.22865· OSTI ID:1883179

^[1]; Glusa, Christian ^[2]

Sandia National Lab. (SNL-CA), Livermore, CA (United States)
Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

Abstract Fractional equations have become the model of choice in several applications where heterogeneities at the microstructure result in anomalous diffusive behavior at the macroscale. In this work we introduce a new fractional operator characterized by a doubly‐variable fractional order and possibly truncated interactions. Under certain conditions on the model parameters and on the regularity of the fractional order we show that the corresponding Poisson problem is well‐posed. We also introduce a finite element discretization and describe an efficient implementation of the finite‐element matrix assembly in the case of piecewise constant fractional order. Through several numerical tests, we illustrate the improved descriptive power of this new operator across media interfaces. Furthermore, we present one‐dimensional and two‐dimensional h ‐convergence results that show that the variable‐order model has the same convergence behavior as the constant‐order model.

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Research Organization:: Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

Sponsoring Organization:: USDOE National Nuclear Security Administration (NNSA); USDOE Laboratory Directed Research and Development (LDRD) Program; USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR)

Grant/Contract Number:: NA0003525; SC0019453; 218318

OSTI ID:: 1883179

Alternate ID(s):: OSTI ID: 1996294

Report Number(s):: SAND2021-10738J; 699117

Journal Information:: Numerical Methods for Partial Differential Equations, Vol. 38, Issue 6; ISSN 0749-159X

Publisher:: WileyCopyright Statement

Country of Publication:: United States

Language:: English

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Title: A fractional model for anomalous diffusion with increased variability: Analysis, algorithms and applications to interface problems

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