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Foundations of Computational Mathematics

, Volume 15, Issue 3, pp 733–791 | Cite as

A PDE Approach to Fractional Diffusion in General Domains: A Priori Error Analysis

  • Ricardo H. Nochetto
  • Enrique Otárola
  • Abner J. Salgado
Article

Abstract

The purpose of this work is to study solution techniques for problems involving fractional powers of symmetric coercive elliptic operators in a bounded domain with Dirichlet boundary conditions. These operators can be realized as the Dirichlet-to-Neumann map for a degenerate/singular elliptic problem posed on a semi-infinite cylinder, which we analyze in the framework of weighted Sobolev spaces. Motivated by the rapid decay of the solution to this problem, we propose a truncation that is suitable for numerical approximation. We discretize this truncation using first degree tensor product finite elements. We derive a priori error estimates in weighted Sobolev spaces. The estimates exhibit optimal regularity but suboptimal order for quasi-uniform meshes. For anisotropic meshes instead, they are quasi-optimal in both order and regularity. We present numerical experiments to illustrate the method’s performance.

Keywords

Fractional diffusion Finite elements Nonlocal operators Degenerate and singular equations Second-order elliptic operators Anisotropic elements 

Mathematics Subject Classification

35S15 65R20 65N12 65N30 

Notes

Acknowledgments

This work is supported by NSF Grants DMS-1109325 and DMS-0807811. A.J.S. is also supported by NSF Grant DMS-1008058 and an AMS-Simons Grant. E.O. is supported by the Conicyt–Fulbright Fellowship Beca Igualdad de Oportunidades.

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Copyright information

© SFoCM 2014

Authors and Affiliations

  • Ricardo H. Nochetto
    • 1
  • Enrique Otárola
    • 2
  • Abner J. Salgado
    • 2
    • 3
  1. 1.Department of Mathematics and Institute for Physical Science and TechnologyUniversity of MarylandCollege ParkUSA
  2. 2.Department of MathematicsUniversity of MarylandCollege ParkUSA
  3. 3.Department of MathematicsUniversity of TennesseKnoxvilleUSA

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