Multi-parameter dimensional split preconditioner for three-by-three block system of linear equations

For a class of three-by-three block systems of linear equations arising from many practical problems, we develop a multi-parameter dimensional split (MPDS) preconditioner to accelerate the convergence of the Krylov subspace methods. Inasmuch as the preconditioning effect of the MPDS preconditioner depends on the values of its parameters, an effective method for computing the optimal parameters is also proposed. Moreover, the eigenvalue distribution of the preconditioned matrix is carefully analyzed. Numerical examples arising from the discretizations of the Navier–Stokes equations and the partial differential equation (PDE) constraint optimization problems are employed to illustrate the robustness and the efficiency of the MPDS preconditioner.