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

TL;DR

This paper introduces a new fractional operator with variable order and truncated interactions to better model anomalous diffusion in heterogeneous media, demonstrating well-posedness, efficient discretization, and improved descriptive power.

Contribution

The paper presents a novel fractional operator with doubly-variable order, analyzes its well-posedness, and develops an efficient finite element method for interface problems.

Findings

The new operator effectively models media interfaces with improved descriptive power.

Finite element discretization achieves convergence similar to constant-order models.

Numerical tests confirm the operator's ability to handle heterogeneities in anomalous diffusion.

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.