An Optimal-Order Numerical Approximation to Variable-order Space-fractional Diffusion Equations on Uniform or Graded Meshes

Abstract

We develop a numerical method for the boundary-value problem of a variable-order linear space-fractional diffusion equation. We prove that if the variable order reduces to an integer at the boundary, then the method discretized on a uniform partition has an optimal-order convergence rate in the $L_\infty$ norm under the smoothness assumption of the data only. Otherwise, the method discretized on a uniform mesh has only a suboptimal-order convergence rate, but the method discretized on a graded mesh has an optimal-order convergence rate in the $L_\infty$ norm assuming the smoothness of data only. Numerical experiments substantiate these theoretical results.