Block Preconditioners Based on Approximate Commutators

This paper introduces a strategy for automatically generating a block preconditioner for solving the incompressible Navier--Stokes equations. We consider the "pressure convection--diffusion preconditioners" proposed by Kay, Loghin, and Wathen [SIAM J. Sci. Comput., 24 (2002), pp. 237-256] and Silvester, Elman, Kay, and Wathen [J. Comput. Appl. Math., 128 (2001), pp. 261-279]. Numerous theoretical and numerical studies have demonstrated mesh independent convergence on several problems and the overall efficacy of this methodology. A drawback, however, is that it requires the construction of a convection--diffusion operator (denoted Fₚ) projected onto the discrete pressure space. This means that integration of this idea into a code that models incompressible flow requires a sophisticated understanding of the discretization and other implementation issues, something often held only by the developers of the model. As an alternative, we consider automatic ways of computing Fₚ based on purely algebraic considerations. The new methods are closely related to the "BFBt preconditioner" of Elman [SIAM J. Sci. Comput., 20 (1999), pp. 1299-1316]. We use the fact that the preconditioner is derived from considerations of commutativity between the gradient and convection--diffusion operators, together with methods for computing sparse approximate inverses, to generate the required matrix Fₚ automatically. We demonstrate that with this strategy the favorable convergence properties of the preconditioning methodology are retained.