A PDE Approach to Numerical Fractional Diffusion

TL;DR

This paper reviews a PDE-based framework for efficiently approximating fractional diffusion problems, emphasizing localization techniques, error analysis, adaptivity, and multilevel methods for nonlocal operators.

Contribution

It introduces a localized PDE approach for fractional diffusion, including error analysis and multilevel methods, applicable to space-time fractional parabolic equations.

Findings

Effective local solution techniques for fractional operators

Rigorous a priori and a posteriori error analyses

Flexible methods for space-time fractional equations

Abstract

Fractional diffusion has become a fundamental tool for the modeling of multiscale and heterogeneous phenomena. However, due to its nonlocal nature, its accurate numerical approximation is delicate. We survey our research program on the design and analysis of efficient solution techniques for problems involving fractional powers of elliptic operators. Starting from a localization PDE result for these operators, we develop local techniques for their solution: a priori and a posteriori error analyses, adaptivity and multilevel methods. We show the flexibility of our approach by proposing and analyzing local solution techniques for a space-time fractional parabolic equation.