Le Borne, Sabine; Oliveira, Suely Joint domain-decomposition \(\mathcal H\)-LU preconditioners for saddle point problems. (English) Zbl 1171.65371 ETNA, Electron. Trans. Numer. Anal. 26, 285-298 (2007). Summary: For saddle point problems in fluid dynamics, several popular preconditioners exploit the block structure of the problem to construct block triangular preconditioners. The performance of such preconditioners depends on whether fast, approximate solvers for the linear systems on the block diagonal (representing convection-diffusion problems) as well as for the Schur complement (in the pressure variables) are available. In this paper, we introduce a completely different approach in which we ignore this given block structure. We instead compute an approximate LU-factorization of the complete system matrix using hierarchical matrix techniques. In particular, we use domain-decomposition clustering with an additional local pivoting strategy to order the complete index set. As a result, we obtain an \(\mathcal H\)-matrix structure in which an \(\mathcal H\)-LU factorization is computed more efficiently and with higher accuracy than for the corresponding block structure based clustering. \(\mathcal H\)-LU preconditioners resulting from the block and joint approaches will be discussed and compared through numerical results. MSC: 65F10 Iterative numerical methods for linear systems 65F30 Other matrix algorithms (MSC2010) 65F35 Numerical computation of matrix norms, conditioning, scaling 65F50 Computational methods for sparse matrices 65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs Keywords:hierarchical matrices; data-sparse approximation; seen equations; preconditioning; saddle point problems; Schur complement; approximate LU-factorization; saddle point problems; Schur complement; approximate LU-factorization; domain-decomposition; numerical results PDFBibTeX XMLCite \textit{S. Le Borne} and \textit{S. Oliveira}, ETNA, Electron. Trans. Numer. Anal. 26, 285--298 (2007; Zbl 1171.65371) Full Text: EuDML EMIS