Numerical solution of nonstationary problems for a space-fractional diffusion equation

TL;DR

This paper develops a finite element method combined with regularized two-level schemes to numerically solve unsteady space-fractional diffusion equations with Robin boundary conditions, demonstrated through 2D numerical experiments.

Contribution

It introduces a novel numerical approach for nonstationary space-fractional diffusion problems using auxiliary pseudo-parabolic equations.

Findings

Effective numerical scheme for space-fractional diffusion

Successful application to 2D model problem

Demonstrated stability and accuracy of the method

Abstract

An unsteady problem is considered for a space-fractional diffusion equation in a bounded domain. A first-order evolutionary equation containing a fractional power of an elliptic operator of second order is studied for general boundary conditions of Robin type. Finite element approximation in space is employed. To construct approximation in time, regularized two-level schemes are used. The numerical implementation is based on solving the equation with the fractional power of the elliptic operator using an auxiliary Cauchy problem for a pseudo-parabolic equation. The results of numerical experiments are presented for a model two-dimensional problem.