A fractional partition of unity finite element method for transient anomalous diffusion problems

Abstract

A fractional partition of unity finite element method is proposed for the solution of the transient anomalous diffusion equation. The Caputo integro-differential operator is employed to represent the fractional time-derivative in these problems. To approximate the Caputo fractional derivative, we propose a new numerical differentiation formula using quadratic splines. For the spatial discretization, we implement an enriched finite element method on unstructured meshes. In the present study, a category of exponential functions incorporating fractional orders is introduced as enrichment functions to refine the finite element approximation. These functions are designed to capture the fractional characteristics of the solution more effectively. By integrating these enrichment functions through the partition of unity framework, the method utilizes prior knowledge of the fractional problem, leading to a substantial enhancement in approximation accuracy while preserving the fundamental advantages of the traditional finite element method. Consequently, the proposed approach delivers precise numerical solutions even with coarse meshes and requires significantly fewer degrees of freedom compared to conventional finite element techniques. Moreover, the mesh resolution remains unaffected by variations in the fractional order, allowing for a consistent mesh structure regardless of changes in fractional parameters. Through extensive numerical simulations, we consistently verify the effectiveness of the proposed technique in achieving high levels of accuracy. This approach not only ensures reliable and precise results but also broadens the applicability of the finite element method, making it more capable of handling time-fractional transient diffusion problems that have traditionally been challenging for standard methods.