On Direct and Semi-Direct Inverse of Stokes, Helmholtz and Laplacian Operators in View of Time-Stepper-Based Newton and Arnoldi Solvers in Incompressible CFD (Communications in Computational Physics, 2013)
Factorization of the incompressible Stokes operator linking pressure and velocity is revisited. The main purpose is to use the inverse of the Stokes operator with alarge time step as a preconditioner for Newton and Arnoldi iterations applied to computation of steady three-dimensional flows and study of their stability. It is shownthat the Stokes operator can be inversed within an acceptable computational effort.This inverse includes fast direct inverses of several Helmholtz operators and iterativeinverse of the pressure matrix. It is shown, additionally, that fast direct solvers can beattractive for the inverse of the Helmholtz and Laplace operators on fine grids and atlarge Reynolds numbers, as well as for other problems where convergence of iterativemethods slows down. Implementation of the Stokes operator inverse to time-stepping-based formulation of the Newton and Arnoldi iterations is discussed.