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Schur complement preconditioners for the Navier–Stokes equations
Author(s) -
Loghin D.,
Wathen A. J.
Publication year - 2002
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.296
Subject(s) - schur complement , preconditioner , generalized minimal residual method , mathematics , saddle point , complement (music) , jacobian matrix and determinant , finite element method , orthogonal complement , navier–stokes equations , iterative method , mathematical optimization , mathematical analysis , geometry , eigenvalues and eigenvectors , compressibility , aerospace engineering , chemistry , engineering , biochemistry , quantum mechanics , subspace topology , thermodynamics , physics , complementation , gene , phenotype
Mixed finite element formulations of fluid flow problems lead to large systems of equations of saddle‐point type for which iterative solution methods are mandatory for reasons of efficiency. A successful approach in the design of solution methods takes into account the structure of the problem; in particular, it is well‐known that an efficient solution can be obtained if the associated Schur complement problem can be solved efficiently and robustly. In this work we present a preconditioner for the Schur complement for the Oseen problem which was introduced in Kay and Loghin (Technical Report 99/06, Oxford University Computing Laboratory, 1999). We show that the spectrum of the preconditioned system is independent of the mesh parameter; moreover, we demonstrate that the number of GMRES iterations grows like the square‐root of the Reynolds number. We also present convergence results for the Schur complement of the Jacobian matrix for the Navier–Stokes operator which exhibit the same mesh independence property and similar growth with the Reynolds number. Copyright © 2002 John Wiley & Sons, Ltd.