### Abstract

We study solution techniques for an evolution equation involving second order derivative in time and the spectral fractional powers, of order s∈ (0 , 1) , of symmetric, coercive, linear, elliptic, second-order operators in bounded domains Ω. We realize fractional diffusion as the Dirichlet-to-Neumann map for a nonuniformly elliptic problem posed on the semi-infinite cylinder C= Ω× (0 , ∞). 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 Ω 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 Ω 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 Ω⊂ R^{2}. We discuss implementation details and report several numerical examples.

Original language | English |
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Pages (from-to) | 177-222 |

Number of pages | 46 |

Journal | Numerische Mathematik |

Volume | 143 |

Issue number | 1 |

Early online date | 25 Jun 2019 |

DOIs | |

Publication status | Published - Sep 2019 |

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### ASJC Scopus subject areas

- Computational Mathematics
- Applied Mathematics

### Cite this

*Numerische Mathematik*,

*143*(1), 177-222. https://doi.org/10.1007/s00211-019-01055-5

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*Numerische Mathematik*, vol. 143, no. 1, pp. 177-222. https://doi.org/10.1007/s00211-019-01055-5

**A PDE approach to fractional diffusion : a space-fractional wave equation.** / Banjai, Lehel; Otárola, Enrique.

Research output: Contribution to journal › Article

TY - JOUR

T1 - A PDE approach to fractional diffusion

T2 - a space-fractional wave equation

AU - Banjai, Lehel

AU - Otárola, Enrique

PY - 2019/9

Y1 - 2019/9

N2 - We study solution techniques for an evolution equation involving second order derivative in time and the spectral fractional powers, of order s∈ (0 , 1) , of symmetric, coercive, linear, elliptic, second-order operators in bounded domains Ω. We realize fractional diffusion as the Dirichlet-to-Neumann map for a nonuniformly elliptic problem posed on the semi-infinite cylinder C= Ω× (0 , ∞). 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 Ω 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 Ω 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 Ω⊂ R2. We discuss implementation details and report several numerical examples.

AB - We study solution techniques for an evolution equation involving second order derivative in time and the spectral fractional powers, of order s∈ (0 , 1) , of symmetric, coercive, linear, elliptic, second-order operators in bounded domains Ω. We realize fractional diffusion as the Dirichlet-to-Neumann map for a nonuniformly elliptic problem posed on the semi-infinite cylinder C= Ω× (0 , ∞). 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 Ω 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 Ω 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 Ω⊂ R2. We discuss implementation details and report several numerical examples.

UR - http://www.scopus.com/inward/record.url?scp=85068161296&partnerID=8YFLogxK

U2 - 10.1007/s00211-019-01055-5

DO - 10.1007/s00211-019-01055-5

M3 - Article

VL - 143

SP - 177

EP - 222

JO - Numerische Mathematik

JF - Numerische Mathematik

SN - 0029-599X

IS - 1

ER -