This paper derives physically meaningful boundary conditions for fractional diffusion equations, using a mass balance approach. Numerical solutions are presented, and theoretical properties are reviewed, including well-posedness and steady state solutions. Absorbing and reflecting boundary conditions are considered, and illustrated through several examples. Reflecting boundary conditions involve fractional derivatives. The Caputo fractional derivative is shown to be unsuitable for modeling fractional diffusion, since the resulting boundary value problem is not positivity preserving.

The numerical approximation of solutions to stochastic partial differential equations with additive spatial white noise on bounded domains in R d is considered. The differential operator is given by the fractional power L β , β ∈ (0, 1), of an integer order elliptic differential operator L and is therefore non-local. Its inverse L −β is represented by a Bochner integral from the Dunford-Taylor functional calculus. By applying a quadrature formula to this integral representation, the inverse fractional power operator L −β is approximated by a weighted sum of non-fractional resolvents (I + t 2 j L) −1 at certain quadrature nodes t j > 0. The resolvents are then discretized in space by a standard finite element method.This approach is combined with an approximation of the white noise, which is based only on the mass matrix of the finite element discretization. In this way, an efficient numerical algorithm for computing samples of the approximate solution is obtained. For the resulting approximation, the strong mean-square error is analyzed and an explicit rate of convergence is derived. Numerical experiments for L = κ 2 − ∆, κ > 0, with homogeneous Dirichlet boundary conditions on the unit cube (0, 1) d in d = 1, 2, 3 spatial dimensions for varying β ∈ (0, 1) attest the theoretical results.where · is the Euclidean norm on R d and Γ, K ν denote the gamma function and the modified Bessel function of the second kind, respectively. Via the positive parameters σ, ν, and κ the most important characteristics of the random field u

Abstract. A unified approach is given for the analysis of the weak error of spatially semidiscrete finite element methods for linear stochastic partial differential equations driven by additive noise. An error representation formula is found in an abstract setting based on the semigroup formulation of stochastic evolution equations. This is then applied to the stochastic heat, linearized Cahn-Hilliard, and wave equations. In all cases it is found that the rate of weak convergence is twice the rate of strong convergence, sometimes up to a logarithmic factor, under the same or, essentially the same, regularity requirements.