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

Lehel Banjai, Enrique Otárola

Research output: Contribution to journalArticle

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⊂ R2. We discuss implementation details and report several numerical examples.

LanguageEnglish
Pages177-222
Number of pages46
JournalNumerische Mathematik
Volume143
Issue number1
Early online date25 Jun 2019
DOIs
Publication statusPublished - Sep 2019

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Fractional Diffusion
Wave equations
Wave equation
Fractional
Hp-FEM
Finite element method
Discretization
Elliptic Problems
Dynamic Boundary Conditions
Dirichlet-to-Neumann Map
Fractional Powers
Evolution Problems
Second-order Derivatives
Stability Estimates
Mesh Refinement
Convex Domain
Exponential Decay
Numerical Approximation
Error Analysis
Truncation

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics

Cite this

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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⊂ R2. We discuss implementation details and report several numerical examples.",
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A PDE approach to fractional diffusion : a space-fractional wave equation. / Banjai, Lehel; Otárola, Enrique.

In: Numerische Mathematik, Vol. 143, No. 1, 09.2019, p. 177-222.

Research output: Contribution to journalArticle

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