Geodesic
On the development of parameter-robust preconditioners and commutator arguments for solving Stokes control problems
Electronic transactions on numerical analysis, Tome 44 (2015), pp. 53-72.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: The development of preconditioners for PDE-constrained optimization problems is a field of numerical analysis which has recently generated much interest. One class of problems which has been investigated in particular is that of Stokes control problems, that is, the problem of minimizing a functional with the Stokes (or Navier-Stokes) equations as constraints. In this manuscript, we present an approach for preconditioning Stokes control problems using preconditioners for the Poisson control problem and, crucially, the application of a commutator argument. This methodology leads to two block diagonal preconditioners for the problem, one of which was previously derived by W. Zulehner in 2011 [SIAM J. Matrix Anal. Appl., 32 (2011), pp. 536 -- 560] using a nonstandard norm argument for this saddle point problem, and the other of which we believe to be new. We also derive two related block triangular preconditioners using the same methodology and present numerical results to demonstrate the performance of the four preconditioners in practice.
Classification : 65F08, 65F10, 65F50, 76D07, 76D55, 93C20
Keywords: PDE-constrained optimization, Stokes control, saddle point system, preconditioning, Schur complement, commutator 
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     author = {Pearson, John W.},
     title = {On the development of parameter-robust preconditioners and commutator arguments for solving {Stokes} control problems},
     journal = {Electronic transactions on numerical analysis},
     pages = {53--72},
     publisher = {mathdoc},
     volume = {44},
     year = {2015},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ETNA_2015__44__a27/}
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Pearson, John W. On the development of parameter-robust preconditioners and commutator arguments for solving Stokes control problems. Electronic transactions on numerical analysis, Tome 44 (2015), pp. 53-72. http://geodesic.mathdoc.fr/item/ETNA_2015__44__a27/