On the efficient preconditioning of the Stokes equations in tight geometries

by   Vladislav Pimanov, et al.

If the Stokes equations are properly discretized, it is well-known that the Schur complement matrix is spectrally equivalent to the identity matrix. Moreover, in the case of simple geometries, it is often observed that most of its eigenvalues are equal to one. These facts form the basis for the famous Uzawa and Krylov-Uzawa algorithms. However, in the case of complex geometries, the Schur complement matrix can become arbitrarily ill-conditioned having a significant portion of non-unit eigenvalues, which makes the established Uzawa preconditioner inefficient. In this article, we study the Schur complement formulation for the staggered finite-difference discretization of the Stokes problem in 3D CT images and synthetic 2D geometries. We numerically investigate the performance of the CG iterative method with the Uzawa and SIMPLE preconditioners and draw several conclusions. First, we show that in the case of low porosity, CG with the SIMPLE preconditioner converges faster to the discrete pressure and provides a more accurate calculation of sample permeability. Second, we show that an increase in the surface-to-volume ratio leads to an increase in the condition number of the Schur complement matrix, while the dependence is inverse for the Schur complement matrix preconditioned with the SIMPLE. As an explanation, we conjecture that the no-slip boundary conditions are the reason for non-unit eigenvalues of the Schur complement.


page 11

page 16


On the structure of the Schur complement matrix for the Stokes equation

In this paper, we investigate the structure of the Schur complement matr...

Computation of transmission eigenvalues by the regularized Schur complement for the boundary integral operators

This paper is devoted to the computation of transmission eigenvalues in ...

Nitsche method for Navier-Stokes equations with slip boundary conditions: Convergence analysis and VMS-LES stabilization

In this paper, we analyze the Nitsche's method for the stationary Navier...

Consistency and Convergence of a High Order Accurate Meshless Method for Solution of Incompressible Fluid Flows

Computations of incompressible flows with velocity boundary conditions r...

A nonlocal Stokes system with volume constraints

In this paper, we introduce a nonlocal model for linear steady Stokes sy...

The nested block preconditioning technique for the incompressible Navier-Stokes equations with emphasis on hemodynamic simulations

We develop a novel iterative solution method for the incompressible Navi...

Please sign up or login with your details

Forgot password? Click here to reset