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

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.

A PDE Approach to Space-Time Fractional Parabolic Problems

“…where Π TY is the interpolation operator defined in [20]. The assertion then follows from the anisotropic interpolation estimates of [20,Theorems 4.7 and 4.8].…”

“…We also define U(T Ω ) := tr Ω V(T Y ), i.e., a P 1 finite element space over the mesh T Ω . The graded meshes described by (4.11) yield near optimal error estimates both in regularity and order for the elliptic case investigated in [20].…”

“…where K s denotes the modified Bessel function of the second kind; see [5,20]. For s ∈ (0, 1), we have [20] 14) and, for a,…”

“…In prior work [20] we used the Caffarelli-Silvestre extension to discretize the fractional space operator and obtained near-optimal error estimates in weighted Sobolev spaces for the extension. We refer the reader to [20] for a an overview of the existing numerical techniques to solve elliptic problems involving fractional diffusion together with their advantages and disadvantages.…”

“…We refer the reader to [20] for a an overview of the existing numerical techniques to solve elliptic problems involving fractional diffusion together with their advantages and disadvantages. In this paper, we will adapt the approach developed in [20] to the parabolic case.…”