A modified improved alternating positive semi-definite splitting preconditioner for double saddle point problems

In this paper, to further enhance the efficiency of the improved alternating positive semi-definite splitting (IAPSS) preconditioner proposed by Ren et al. (Numer Algorithms 91:1363–1379, 2022. https://doi.org/10.1007/s11075-022-01305-y), the modified IAPSS preconditioner is established, which can be applied to GMRES method to solve the double saddle point problems. The construction idea of the preconditioner is to modify several sub-matrices in the IAPSS preconditioner. Theoretically, the iteration method generated by the proposed preconditioner is unconditionally convergent for all positive parameters. Furthermore, the selection of the parameters is discussed in detail. Finally, the performance of the preconditioner is verified by the two examples of the liquid crystal director model and the mixed Stokes/Darcy model.