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

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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.

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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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  1. Abner J. Salgado

    Present address: Department of Mathematics, University of Tennesse, Knoxville, TN, 37996, USA

Authors and Affiliations

  1. Department of Mathematics and Institute for Physical Science and Technology, University of Maryland, College Park, MD, 20742, USA

    Ricardo H. Nochetto

  2. Department of Mathematics, University of Maryland, College Park, MD, 20742, USA

    Enrique Otárola & Abner J. Salgado

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  1. Ricardo H. Nochetto
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Correspondence to Ricardo H. Nochetto.

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Communicated by Albert Cohen.

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Nochetto, R.H., Otárola, E. & Salgado, A.J. A PDE Approach to Fractional Diffusion in General Domains: A Priori Error Analysis. Found Comput Math 15, 733–791 (2015). https://doi.org/10.1007/s10208-014-9208-x

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