A symbol-based preconditioner for a sixth-order scheme from multi-dimensional steady-state Riesz space fractional diffusion equations

In this paper, we employ a sixth-order numerical scheme to approximate the multi-dimensional steady-state Riesz space-fractional diffusion equations (RSFDEs) and subsequently propose a preconditioned conjugate gradient (PCG) method with a symbol-based preconditioner for solving the resulting linear systems. Theoretically, we prove that the PCG solver achieves an optimal convergence rate — i.e., a convergence rate independent of discretization step size — by showing that the spectra of the preconditioned matrices are uniformly bounded within the open interval . Numerical experiments validate the effectiveness of the proposed preconditioner for three-dimensional steady-state RSFDEs and confirm the rapid convergence of the PCG method.