Analysis and numerical approximation for space-dependent variable-order fractional diffusion equation in the flowing media

Abstract We investigate a fully discrete scheme for a space-dependent variable-order fractional diffusion equation in the flowing media, which can be derived by introducing a velocity field to continuous time random walk model with waiting time obeying a spatially dependent power-law distribution. We provide regularity estimates for the solution under some regularity assumptions on the variable-order αx and the velocity field v. A temporal semidiscrete scheme generated by the backward Euler convolution quadrature method is proposed, and an Oτ convergence rate is obtained by some skillful error analyses. Then, the fully discrete scheme is built by using finite element method to approximate the spatial operator, and an optimal spatial error estimate is obtained by introducing some discrete operators, i.e., the convergence order can well match the order of optimal spatial regularity of the solution. Finally, various numerical examples are presented to validate our theoretical results.