On preconditioning incompressible non-Newtonian flow problems

This paper concerns preconditioned iterative solution methods for solving incompressible non-Newtonian Navier-Stokes equations as arising in regularized Bingham models. Stable finite element discretization applied to the linearized equations results in linear systems of saddle point form. In order to circumvent the difficulties of efficiently and cheaply preconditioning the Schur complement of the system matrix, in this paper the augmented Lagrangian (AL) technique is used to algebraically transform the original system into an equivalent one, which does not change the solution and that is the linear system we intend to solve by some preconditioned iterative method. For the transformed system matrix a lower block-triangular preconditioner is proposed. The crucial point in the AL technique is how to efficiently solve the modified pivot block involving the velocity. In this paper an approximation of the modified pivot block is proposed and an algebraic multi-grid technique is applied for this approximation. Numerical experiments show that the AL preconditioner combining with the algebraic multi-grid solver is quite efficient and robust with respect to the variation of the mesh size and the regularized parameter of the Bingham model. Mathematics subject classification: 65F10 65F08 65N30.