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 $F_p$) 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 $F_p$ 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 $F_p$ automatically. We demonstrate that with this strategy the favorable convergence properties of the preconditioning methodology are retained.