Multigrid method with greedy partial block Jacobi smoother for solving two-dimensional space-fractional diffusion equations

Based on the block Jacobi splitting, a kind of greedy partial block Jacobi (GPBJ) iteration method is constructed by greedily selecting the blocks with relatively large residuals and performing the block Jacobi iteration on the selected blocks. Theoretical analysis demonstrates that the GPBJ iteration is unconditionally convergent if the coefficient matrix of the linear system is H-matrix. Then combining with the alternating direction strategy, the GPBJ smoothed multigrid method is designed to solve the discrete linear system of two-dimensional space-fractional diffusion equations, where the coefficient matrix is strictly diagonally dominant. Numerical experiments indicate that the multigrid method smoothed by the GPBJ iteration can significantly reduce the computation time for solving the considered discrete linear system.