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Efficient preconditioning techniques for finite‐element quadratic discretization arising from linearized incompressible Navier–Stokes equations
Author(s) -
El Maliki A.,
Guénette R.
Publication year - 2010
Publication title -
international journal for numerical methods in fluids
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.938
H-Index - 112
eISSN - 1097-0363
pISSN - 0271-2091
DOI - 10.1002/fld.2136
Subject(s) - preconditioner , mathematics , discretization , schur complement , navier–stokes equations , finite element method , saddle point , mathematical analysis , compressibility , linear system , geometry , physics , thermodynamics , eigenvalues and eigenvectors , quantum mechanics
We develop an efficient preconditioning techniques for the solution of large linearized stationary and non‐stationary incompressible Navier–Stokes equations. These equations are linearized by the Picard and Newton methods, and linear extrapolation schemes in the non‐stationary case. The time discretization procedure uses the Gear scheme and the second‐order Taylor–Hood element P 2 − P 1 is used for the approximation of the velocity and the pressure. Our purpose is to develop an efficient preconditioner for saddle point systems. Our tools are the addition of stabilization (penalization) term r ∇(div(·)), and the use of triangular block matrix as global preconditioner. This preconditioner involves the solution of two subsystems associated, respectively, with the velocity and the pressure and have to be solved efficiently. Furthermore, we use the P 1 − P 2 hierarchical preconditioner recently proposed by the authors, for the block matrix associated with the velocity and an additive approach for the Schur complement approximation. Finally, several numerical examples illustrating the good performance of the preconditioning techniques are presented. Copyright © 2009 John Wiley & Sons, Ltd.