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

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

Journal Article · 06 February 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)

Fractional equations have become the model of choice in several applications where heterogeneities at the microstructure result in anomalous diffusive behavior at the macroscale. Here, 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. Additionally, we 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.

View Accepted Manuscript (DOE)

Research Organization:: Sandia National Laboratories (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

OSTI ID:: 1883179

Alternate ID(s):: OSTI ID: 1996294

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

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

Publisher:: WileyCopyright Statement

Country of Publication:: United States

Language:: English

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