Numerische Mathematik

, Volume 143, Issue 1, pp 177–222 | Cite as

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

  • Lehel Banjai
  • Enrique OtárolaEmail author


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.

Mathematics Subject Classification

26A33 35J70 35R11 65M12 65M15 65M60 



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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Departamento de MatemáticaUniversidad Técnica Federico Santa MaríaValparaisoChile
  2. 2.Maxwell Institute for Mathematical Sciences, School of Mathematical and Computer SciencesHeriot-Watt UniversityEdinburghUK

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