Numerical algorithm for the model describing anomalous diffusion in expanding media

TL;DR

This paper introduces a finite element and convolution quadrature-based numerical algorithm for simulating anomalous diffusion in expanding media, with proven error estimates and verified effectiveness through numerical experiments.

Contribution

It develops a new numerical scheme combining finite element and convolution quadrature methods for a fractional diffusion model in expanding media, with rigorous error analysis.

Findings

Error estimates for semi-discrete and fully discrete schemes

Numerical experiments confirm the algorithm's effectiveness

Sobolev regularity of the solution is established

Abstract

We provide a numerical algorithm for the model characterizing anomalous diffusion in expanding media, which is derived in [F. Le Vot, E. Abad, and S. B. Yuste, Phys. Rev. E {\bf96} (2017) 032117]. The Sobolev regularity for the equation is first established. Then we use the finite element method to discretize the Laplace operator and present error estimate of the spatial semi-discrete scheme based on the regularity of the solution; the backward Euler convolution quadrature is developed to approximate Riemann-Liouville fractional derivative and error estimates for the fully discrete scheme are established by using the continuity of solution. Finally, the numerical experiments verify the effectiveness of the algorithm.