On the efficient preconditioning of the Stokes equations in tight geometries

TL;DR

This paper investigates the limitations of the Uzawa preconditioner for the Stokes equations in complex geometries and proposes the SIMPLE preconditioner as a more effective alternative, improving computational efficiency and accuracy.

Contribution

It introduces the SIMPLE preconditioner tailored for tight geometries, addressing the ill-conditioning issues of the Schur complement in such cases.

Findings

SIMPLE preconditioner outperforms Uzawa in complex geometries.

Eigenvalues of the Schur complement deviate from one due to boundary conditions.

SIMPLE improves permeability calculations in tight porous media.

Abstract

If the Stokes equations are properly discretized, it is 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 algorithm. Despite recent progress in developing efficient iterative methods for solving the Stokes problem, the Uzawa algorithm remains popular in science and engineering, especially when accelerated by Krylov subspace methods. However, in complex geometries, the Schur complement matrix can become severely ill-conditioned, having a significant portion of non-unit eigenvalues. This makes the established Uzawa preconditioner inefficient. To explain this behaviour, we examine the Pressure Schur Complement formulation for the staggered finite-difference discretization of the Stokes equations. Firstly, we conjecture that the no-slip boundary conditions are the reason for non-unit eigenvalues of the Schur complement matrix. Secondly, we demonstrate that its condition number increases with increasing the surface-to-volume ratio of the flow domain. As an alternative to the Uzawa preconditioner, we propose using the diffusive SIMPLE preconditioner for geometries with a large surface-to-volume ratio. We show that the latter is much more fast and robust for such geometries. Furthermore, we show that the usage of the SIMPLE preconditioner leads to more accurate practical computation of the permeability of tight porous media. Keywords: Stokes problem, tight geometries, computing permeability, preconditioned Krylov subspace methods