A semi-discrete finite element method for a class of time-fractional diffusion equations

As fractional diffusion equations can describe the early breakthrough and the heavy-tail decay features observed in anomalous transport of contaminants in groundwater and porous soil, they have been commonly used in the related mathematical descriptions. These models usually involve long-time-range computation, which is a critical obstacle for their application; improvement of computational efficiency is of great significance. In this paper, a semi-discrete method is presented for solving a class of time-fractional diffusion equations that overcome the critical long-time-range computation problem. In the procedure, the spatial domain is discretized by the finite element method, which reduces the fractional diffusion equations to approximate fractional relaxation equations. As analytical solutions exist for the latter equations, the burden arising from long-time-range computation can effectively be minimized. To illustrate its efficiency and simplicity, four examples are presented. In addition, the method is used to solve the time-fractional advection–diffusion equation characterizing the bromide transport process in a fractured granite aquifer. The prediction closely agrees with the experimental data, and the heavy-tail decay of the anomalous transport process is well represented.

A Novel Finite Element Method for a Class of Time Fractional Diffusion Equations

Anomalous transport of contaminants in groundwater or porous soil is a research focus in hydrology and soil science for decades. Because fractional diffusion equations can well characterize early breakthrough and heavy tail decay features of contaminant transport process, they have been considered as promising tools to simulate anomalous transport processes in complex media. However, the efficient and accurate computation of fractional diffusion equations is a main task in their applications. In this paper, we introduce a novel numerical method which captures the critical Mittag-Leffler decay feature of subdiffusion in time direction, to solve a class of time fractional diffusion equations. A key advantage of the new method is that it overcomes the critical problem in the application of time fractional differential equations: long-time range computation. To illustrate its efficiency and simplicity, three typical academic examples are presented. Numerical results show a good agreement with the exact solutions.