A Unified View of Some Numerical Methods for Fractional Diffusion

Dr. Clemens Hofreither

May 28, 2019, 3:30 p.m. S2 416-1

In recent years, a number of numerical methods for the solution of
fractional Laplace and, more generally, fractional diffusion problems
have been proposed. The approaches are quite diverse and include, among
others, the use of best uniform rational approximations, quadrature for
Dunford-Taylor-like integrals, finite element approaches for a localized
elliptic extension into a space of increased dimensions, and time
stepping methods for a parabolic reformulation of the fractional
differential equation. We review these methods and observe that all
approaches mentioned above can, in fact, be interpreted as realizing
different rational approximations of a univariate function over the
spectrum of the original (non-fractional) diffusion operator.

This observation allows us to cast all described methods into a unified
theoretical and computational framework, which has a number of benefits.
Theoretically, it enables us to give new convergence proofs for several
of the studied methods, clarifies similarities and differences between
the approaches, suggests how to design new and improved methods, and
allows a direct comparison of the relative performance of the various
methods. Practically, it provides a single, simple to implement,
efficient and fully parallel algorithm for the realization of all
studied methods; for instance, this does away with the need for
constructing specific multilevel methods for the efficient realization
of the extension methods and lets us parallelize the otherwise
inherently sequential time stepping approach.

In a detailed numerical study, we compare all investigated methods for
various fractional exponents and draw conclusions from the results.