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

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 $$\varOmega $$ . We realize fractional diffusion as the Dirichlet-to-Neumann map for a nonuniformly elliptic problem posed on the semi-infinite cylinder $$\mathcal {C}= \varOmega \times (0,\infty )$$ . 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: the trapezoidal and leapfrog schemes. The discretization in space relies on the tensorization of a first-degree FEM in $$\varOmega $$ with a suitable hp-FEM in the extended variable. For both schemes we derive stability and error estimates. We consider a first-degree FEM in $$\varOmega $$ with mesh refinement near corners and the aforementioned hp-FEM in the extended variable and extend the a priori error analysis of the trapezoidal scheme for open, bounded, polytopal but not necessarily convex domains $$\varOmega \subset {\mathbb {R}}^2$$ . We discuss implementation details and report several numerical examples.