A Pde Approach to Fractional Diffusion in General Domains: a Priori

Abstract

Abstract. The purpose of this work is the study of 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 of 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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Cite this paper

@inproceedings{ANALYSIS2014APA, title={A Pde Approach to Fractional Diffusion in General Domains: a Priori}, author={ERROR ANALYSIS and Ricardo H. Nochetto and Abner J. Salgado and Albert Cohen}, year={2014} }