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

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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.

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Correspondence to Enrique Otárola.

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EO is partially supported by CONICYT through FONDECYT Project 11180193.

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Banjai, L., Otárola, E. A PDE approach to fractional diffusion: a space-fractional wave equation. Numer. Math. 143, 177–222 (2019).

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Mathematics Subject Classification

  • 26A33
  • 35J70
  • 35R11
  • 65M12
  • 65M15
  • 65M60