TY - JOUR
T1 - Block preconditioners based on approximate commutators
AU - Elman, Howard
AU - Howle, Victoria E.
AU - Shadid, John
AU - Shuttleworth, Robert
AU - Tuminaro, Ray
PY - 2006
Y1 - 2006
N2 - 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 F p automatically. We demonstrate that with this strategy the favorable convergence properties of the preconditioning methodology are retained.
AB - 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 F p automatically. We demonstrate that with this strategy the favorable convergence properties of the preconditioning methodology are retained.
KW - Iterative algorithms
KW - Navier-Stokes
KW - Preconditioning
UR - http://www.scopus.com/inward/record.url?scp=33748791765&partnerID=8YFLogxK
U2 - 10.1137/040608817
DO - 10.1137/040608817
M3 - Article
AN - SCOPUS:33748791765
SN - 1064-8275
VL - 27
SP - 1651
EP - 1668
JO - SIAM Journal on Scientific Computing
JF - SIAM Journal on Scientific Computing
IS - 5
ER -