This paper is concerned with the fast algorithm for solving multidimensional spatial fractional Cahn-Hilliard equations. The equations are discretized by a linear and energy-stable finite difference scheme. It gives a system of linear equations with a \(2\times 2\) indefinite ill-conditioned block matrix. We construct a positive definite block preconditioner based on the sine transform for the minimal residual method to solve the indefinite system. Theoretically, we prove that all the eigenvalues of the preconditioned matrix are located in the intervals \([-\frac{3}{2},-\frac{1}{2\sqrt{2}}]\cup [\frac{1}{2\sqrt{2}},\frac{3}{2}]\) without outliers. Thus, the preconditioned MINRES method has a linear convergence rate within an iteration number independent of the matrix size. A fast implementation is presented for the preconditioned matrix–vector multiplication, which reduces the computation complexity significantly. The matrix-size independent convergence rate and the fast implementation guarantee a linearithmic (nearly optimal) complexity of the proposed solver. Numerical examples are given to demonstrate the effectiveness of the proposed method.
