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A preconditioner for generalized saddle point problems: application to 3D stationary Navier-Stokes equations

Abstract : In this article we consider the stationary Navier‐Stokes system discretized by finite element methods which do not satisfy the inf‐sup condition. These discretizations typically take the form of a variational problem with stabilization terms. Such a problem may be transformed by iteration methods into a sequence of linear, Oseen‐type variational problems. On the algebraic level, these problems belong to a certain class of linear systems with nonsymmetric system matrices (“generalized saddle point problems”). We show that if the underlying finite element spaces satisfy a generalized inf‐sup condition, these problems have a unique solution. Moreover, we introduce a block triangular preconditioner and we show how the eigenvalue bounds of the preconditioned system matrix depend on the coercivity constant and continuity bounds of the bilinear forms arising in the variational problem. Finally we prove that the stabilized P1‐P1 finite element method proposed by Rebollo is covered by our theory and we show that the condition number of the preconditioned system matrix is independent of the mesh size. Numerical tests with 3D stationary Navier‐Stokes flows confirm our results.
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https://hal.archives-ouvertes.fr/hal-00768482
Contributor : Caterina Calgaro Connect in order to contact the contributor
Submitted on : Friday, December 21, 2012 - 3:55:44 PM
Last modification on : Wednesday, March 23, 2022 - 3:50:58 PM

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Caterina Calgaro, Paul Deuring, Delphine Jennequin. A preconditioner for generalized saddle point problems: application to 3D stationary Navier-Stokes equations. Numerical Methods for Partial Differential Equations, Wiley, 2006, 22 (6), pp.1289--1313. ⟨10.1002/num.20154⟩. ⟨hal-00768482⟩

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