A PDE Approach to Space-Time Fractional Parabolic Problems

Abstract

Abstract. We study solution techniques for parabolic equations with fractional diffusion and Caputo fractional time derivative, the latter being discretized and analyzed in a general Hilbert space setting. The spatial fractional diffusion is realized as the Dirichlet-to-Neumann map for a nonuniformly elliptic problem posed on a semi-infinite cylinder in one more spatial dimension. We write our evolution problem as a quasi-stationary elliptic problem with a dynamic boundary condition. We propose and analyze an implicit fully-discrete scheme: first-degree tensor product finite elements in space and an implicit finite difference discretization in time. We prove stability and error estimates for this scheme.

A FEM for an optimal control problem subject to the fractional Laplace equation

“…[21,Section 5.1]. A convergence rate of h 1+s (up to some logarithmic term) in the L 2 (Ω)-norm can be obtained [1,22], provided that z ∈ H 1−s (Ω), where h denotes the global mesh parameter. However, numerical experiments show that this convergence rate is not optimal in a specific range of fractional powers s. The cost of solving the problem is related to the number of elements in C Y , and not only to the number of elements in Ω, resulting in an increased computational complexity.…”

Analysis of an approximation to a fractional extension problem

“…Due to its wide array of applications, the computation of fractional powers of the Laplacian, and other elliptic operators, has become a problem of great interest [2,9,25,26]. Fractional powers of operators have received this attention due to their applicability to the accurate modeling of real-world problems with varying scales.…”

“…Further details and applications of this method may be found in [3,7,14,31]. Several numerical algorithms for solving problems involving fractional powers of the Laplacian have been developed based upon the Caffarelli-Silvestre extension technique [12,18,25,26]. These methods have also been extended to problems involving general elliptic operators and various other physically relevant nonlocal problems.…”

A PDE approach to fractional diffusion: a space-fractional wave equation

“…The design of efficient solution techniques for problems involving fractional diffusion is intricate, mainly due to the nonlocal character of L s [7,8,9,12]. Recently, and in order to overcome such a nonlocal feature, it is has been proved useful in numerical analysis the application of the Caffarelli-Silvestre extension [4,41,40]. When L = −∆ and Ω = R n , i.e., in the case of the Laplacian in the whole space, Caffarelli and Silvestre [9] showed that L s can be realized as the Dirichlet-to-Neumann map for an extension problem to the upper half-space R n+1 + ; the extension corresponds to a nonuniformly elliptic PDE.…”