A PDE approach to fractional diffusion: a space-fractional wave equation

TL;DR

This paper develops a PDE-based framework for solving space-fractional wave equations, introducing a novel elliptic problem formulation, regularity analysis, and two stable, error-estimated numerical schemes for efficient approximation.

Contribution

It presents a new PDE approach to fractional diffusion, reformulating the problem as a quasi-stationary elliptic problem with dynamic boundary conditions, and proposes two fully discrete schemes with stability and error analysis.

Findings

The extended problem exhibits exponential decay, enabling effective truncation for numerical methods.

Two fully discrete schemes are developed with proven stability and error estimates.

The approach provides a rigorous framework for space-fractional wave equations with regularity and approximation guarantees.

Abstract

We study solution techniques for an evolution equation involving second order derivative in time and the spectral fractional powers, of order $s \in (0,1)$, of symmetric, coercive, linear, elliptic, second-order operators in bounded domains $\Omega$. We realize fractional diffusion as the Dirichlet-to-Neumann map for a nonuniformly elliptic problem posed on the semi-infinite cylinder $\mathcal{C} = \Omega \times (0,\infty)$. We thus rewrite our evolution problem as a quasi-stationary elliptic problem with a dynamic boundary condition and derive space, time, and space-time regularity estimates for its solution. The latter problem exhibits an exponential decay in the extended dimension and thus suggests a truncation that is suitable for numerical approximation. We propose and analyze two fully discrete schemes. The discretization in time is based on finite difference discretization techniques: trapezoidal and leapfrog schemes. The discretization in space relies on the tensorization of a first-degree FEM in $\Omega$ with a suitable $hp$-FEM in the extended variable. For both schemes we derive stability and error estimates.