Two Fully Discrete Schemes for Fractional Diffusion and Diffusion-Wave Equations with Nonsmooth Data

Bangti Jin, Raytcho Lazarov, Zhi Zhou
2016 SIAM Journal on Scientific Computing  
We consider initial/boundary value problems for the subdiffusion and diffusionwave equations involving a Caputo fractional derivative in time. We develop two fully discrete schemes based on the piecewise linear Galerkin finite element method in space and convolution quadrature in time with the generating function given by the backward Euler method/second-order backward difference method, and establish error estimates optimal with respect to the regularity of problem data. These two schemes are
more » ... irst and second-order accurate in time for both smooth and nonsmooth data. Extensive numerical experiments for two-dimensional problems confirm the convergence analysis and robustness of the schemes with respect to data regularity. Note that if α = 1 and α = 2, then equation (1.1) represents a parabolic and a hyperbolic equation, respectively. In this paper we focus on the fractional cases 0 < α < 1 and 1 < α < 2, with a Caputo derivative, which are known as the subdiffusion and diffusion-wave equation, respectively, in the literature. In analogy with Brownian motion for normal diffusion, the model (1.1) with 0 < α < 1 is the macroscopic counterpart of continuous time random walk [38, 13] . Throughout, equation (1.1) is subjected to the following boundary condition u(x, t) = 0, (x, t) ∈ ∂Ω × (0, T ),
doi:10.1137/140979563 fatcat:odijwwnzavbtvcykhth25hp6qq