On the iterative solution of the Stokes problem

Block preconditioned iterative conjugate gradient methods for solving the three-dimensional Stokes problem are investigated. A formulation with a computational domain in the form of a parallelepiped is considered. “Standard” approximations on a cubic staggered grid, seven-point and two-point for the Laplace and derivative operators are used. The resulting saddle-type SLAE is regularized to ensure the uniqueness of the solution. The block preconditioner is constructed by the method of incomplete factorization with diagonal compensation and using band approximations for matrices inverse to the Schur complement and the grid Laplace operator, as well as using an algebraic multigrid algorithm. Examples of numerical experiments on a representative series of methodological problems using parallel algorithms on different numbers of processors are given. The issues of generalizing the proposed approaches to broader classes of problems are considered.