A fractional model for anomalous diffusion with increased variability: Analysis, algorithms and applications to interface problems. Issue 6 (7th February 2022)

• Record Type: Journal Article
• Authors: D'Elia, Marta; Glusa, Christian
• Other Names: Bochev Pavel guestEditor.; D'Elia Marta guestEditor.; Du Qiang guestEditor.; Hou Steve guestEditor.; Webster Clayton guestEditor.; Zhang Guannan guestEditor.
• Is Part Of: Numerical methods for partial differential equations. Volume 38:Issue 6(2022)
• Page Start: 2084
• Page End: 2103
• Publication Date: 2022-02-07
• Subjects: anomalous diffusion -- interface problems -- subsurface diffusion -- variable‐order fractional operators; Differential equations, Partial -- Numerical solutions -- Periodicals; 515.353
• DOI: 10.1002/num.22865
• Languages: English
• ISSNs: 0749-159X

Abstract

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.