A robust two-level incomplete factorization for (Navier-)Stokes saddle point matrices

TL;DR

This paper introduces a robust two-level hybrid direct/iterative method for solving saddle point matrices from Navier-Stokes equations, offering stability, scalability, and applicability to related problems like electrical network simulations.

Contribution

A novel two-level approach that is robust, parameter-controlled, scalable, and applicable recursively to saddle point matrices from fluid dynamics and electrical network problems.

Findings

Method is robust near instability points

Convergence rate is independent of problem size

Applicable to Poisson and electrical network problems

Abstract

We present a new hybrid direct/iterative approach to the solution of a special class of saddle point matrices arising from the discretization of the steady incompressible Navier-Stokes equations on an Arakawa C-grid. The two-level method introduced here has the following properties: (i) it is very robust, even close to the point where the solution becomes unstable; (ii) a single parameter controls fill and convergence, making the method straightforward to use; (iii) the convergence rate is independent of the number of unknowns; (iv) it can be implemented on distributed memory machines in a natural way; (v) the matrix on the second level has the same structure and numerical properties as the original problem, so the method can be applied recursively; (vi) the iteration takes place in the divergence- free space, so the method qualifies as a 'constraint preconditioner'; (vii) the approach can also be applied to Poisson problems. This work is also relevant for problems in which similar saddle point matrices occur, for instance when simulating electrical networks, where one has to satisfy Kirchhoff's conservation law for currents.