2021/01/27 by Marta D’Elia, D'Elia, Marta, Christian Glusa +1
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Differential Equations and Numerical Methods #FOS: Mathematics #Fractional Differential Equations Solutions #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.2101.11765
openalex publication_date 2021/01/27 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
Fractional equations have become the model of choice in several applications\nwhere heterogeneities at the microstructure result in anomalous diffusive\nbehavior at the macroscale. In this work we introduce a new fractional operator\ncharacterized by a doubly-variable fractional order and possibly truncated\ninteractions. Under certain conditions on the model parameters and on the\nregularity of the fractional order we show that the corresponding Poisson\nproblem is well-posed. We also introduce a finite element discretization and\ndescribe an efficient implementation of the finite-element matrix assembly in\nthe case of piecewise constant fractional order. Through several numerical\ntests, we illustrate the improved descriptive power of this new operator across\nmedia interfaces. Furthermore, we present one-dimensional and two-dimensional\nh-convergence results that show that the variable-order model has the same\nconvergence behavior as the constant-order model.\n