Tensor FEM for Spectral Fractional Diffusion

Abstract

We design and analyze several Finite Element Methods (FEMs) applied to the Caffarelli-Silvestre extension that localizes the fractional powers of symmetric, coercive, linear elliptic operators in bounded domains with Dirichlet boundary conditions. We consider open, bounded, polytopal but not necessarily convex domains Ω ⊂ R d with d = 1, 2. For the solution to the extension problem, we establish analytic regularity with respect to the extended variable y ∈ (0, ∞). We prove that the solution belongs to countably normed, power-exponentially weighted Bochner spaces of analytic functions with respect to y, taking values in corner-weighted Kondat'ev type Sobolev spaces in Ω. In Ω ⊂ R 2 , we discretize with continuous, piecewise linear, Lagrangian FEM (P 1 -FEM) with mesh refinement near corners, and prove that first order convergence rate is attained for compatible data f ∈ H 1−s (Ω).We also prove that tensorization of a P 1 -FEM in Ω with a suitable hp-FEM in the extended variable achieves log-linear complexity with respect to N Ω , the number of degrees of freedom in the domain Ω. In addition, we propose a novel, sparse tensor product FEM based on a multilevel P 1 -FEM in Ω and on a P 1 -FEM on radical-geometric meshes in the extended variable. We prove that this approach also achieves log-linear complexity with respect to N Ω . Finally, under the stronger assumption that the data is analytic in Ω, and without compatibility at ∂Ω, we establish exponential rates of convergence of hp-FEM for spectral, fractional diffusion operators in energy norm. This is achieved by a combined tensor product hp-FEM for the Caffarelli-Silvestre extension in the truncated cylinder Ω × (0, Y ) with anisotropic geometric meshes that are refined towards ∂Ω. We also report numerical experiments for model problems which confirm the theoretical results. We indicate several extensions and generalizations of the proposed methods to other problem classes and to other boundary conditions on ∂Ω.

An adaptive finite element method for the sparse optimal control of fractional diffusion

“…We refer the reader to ( [5], Remark 2.1) for a discussion. The second feature is the lack of boundary regularity [9], which leads to reduced convergence rates [10,11]. The first, and most important, is that (−Δ) s is a nonlocal operator [6][7][8].…”

“…With the extension at hand, we thus introduce the fundamental result by Caffarelli and Silvestre [6][7][8]: the Dirichlet-to-Neumann map of problem (1.7) and the spectral fractional Laplacian are related by s (−Δ) s u = in Ω. Such an interpolation theory allows for tensor product elements that exhibit an anisotropic feature in the extended dimension, that is in turn needed to compensate the singular behavior of the solution in the extended variable y ( [11], Theorem 2.7, [10], Theorem 4.7]). The main advantage of the scheme proposed in [11] is that it involves the resolution of the local problem (1.7) and thus its implementation uses basic ingredients of finite element analysis; its analysis, however, involves asymptotic estimates of Bessel functions [13], to derive regularity estimates in weighted Sobolev spaces, elements of harmonic analysis [14,15], and an anisotropic polynomial interpolation theory in weighted Sobolev spaces [16,17].…”

“…The first, and most important, is that (−Δ) s is a nonlocal operator [6][7][8]. The second feature is the lack of boundary regularity [9], which leads to reduced convergence rates [10,11]. In fact, as ( [9], Theorem 1.3) shows, if Ω is sufficiently smooth, then the solution u to problem (1.3) behaves like…”

“…The main advantage of the scheme proposed in [11] is that it involves the resolution of the local problem (1.7) and thus its implementation uses basic ingredients of finite element analysis; its analysis, however, involves asymptotic estimates of Bessel functions [13], to derive regularity estimates in weighted Sobolev spaces, elements of harmonic analysis [14,15], and an anisotropic polynomial interpolation theory in weighted Sobolev spaces [16,17]. Such an interpolation theory allows for tensor product elements that exhibit an anisotropic feature in the extended dimension, that is in turn needed to compensate the singular behavior of the solution in the extended variable y ( [11], Theorem 2.7, [10], Theorem 4.7]).…”

A reaction coefficient identification problem for fractional diffusion

“…We derive approximation properties of its solution. In section 5.1 we briefly recall the finite element scheme of [2] that approximates the solution to (1.7). In section 5.2 we introduce a fully discrete scheme for the truncated identification problem and derive convergence of discrete solutions when the regularization parameter converges to zero.…”