Ku, Jaeun; Reichel, Lothar Simple efficient solvers for certain ill-conditioned systems of linear equations, including \(H(\operatorname{div})\) problems. (English) Zbl 1457.65163 J. Comput. Appl. Math. 343, 240-249 (2018). Summary: Simple, accurate, and efficient iterative methods for the solution of a nearly singular variational problem are described. The systems considered arise, e.g., when seeking to determine the flux of second order elliptic partial differential equations. Each (outer) iteration uses a robust and efficient solver, which may be a direct or iterative method, and only a few (outer) iterations are required to obtain an approximate solution. Reduced rank extrapolation may be applied to speed up the convergence. Cited in 3 Documents MSC: 65N20 Numerical methods for ill-posed problems for boundary value problems involving PDEs 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 65N22 Numerical solution of discretized equations for boundary value problems involving PDEs 35A15 Variational methods applied to PDEs Keywords:finite element method; nearly singular system; variational problem Software:DistMesh; PCBDDC; mfem; AGMG PDFBibTeX XMLCite \textit{J. Ku} and \textit{L. Reichel}, J. Comput. Appl. Math. 343, 240--249 (2018; Zbl 1457.65163) Full Text: DOI References: [1] Arnold, D. N.; Falk, R. S.; Winther, R., Preconditioning in \(H(d i v)\) and applications, Math. Comp., 66, 957-984, (1997) · Zbl 0870.65112 [2] Hiptmair, R.; Xu, J., Nodal auxiliary space preconditioning in \(H(c u r l)\) and \(H(d i v)\) spaces, SIAM J. Numer. Anal., 45, 2483-2509, (2007) · Zbl 1153.78006 [3] Ku, J.; Lee, Y. J.; Sheen, D., A hybrid two-step finite element method for flux approximation: a priori and a posteriori estimates, ESAIM Math. Model. Numer. Anal. (ESAIM: M2AN), 51, 1303-1316, (2017) · Zbl 1379.65089 [4] Brown, P. N.; Walker, H. F., GMRES on (nearly) singular systems, SIAM J. Matrix Anal. Appl., 18, 37-51, (1997) · Zbl 0876.65019 [5] Eiermann, M.; Marek, I.; Niethammer, W., On the solution of singular linear systems of algebraic equations by semiiterative methods, Numer. Math., 53, 265-283, (1988) · Zbl 0655.65049 [6] Hager, W. W., Iterative methods for nearly singular linear systems, SIAM J. Sci. Comput., 22, 747-766, (2000) · Zbl 0967.65043 [7] Hayami, K.; Sugihara, M., A geometric view of Krylov subspace methods on singular systems, Numer. Linear Algebra Appl., 18, 449-460, (2011), and 21 (2014), pp. 701-702 · Zbl 1245.65037 [8] Reichel, L.; Ye, Q., Breakdown-free GMRES for singular systems, SIAM J. Matrix Anal. Appl., 26, 1001-1021, (2005) · Zbl 1086.65030 [9] Zampini, S., PCBDDC: a class of robust dual-primal methods in petsc, SIAM J. Sci. Comput., 38, S282-S306, (2016) · Zbl 1352.65632 [10] Brezinski, C.; Redivo Zaglia, M., Extrapolation methods: theory and practice, (1991), North-Holland Amsterdam · Zbl 0744.65004 [11] Duminil, S.; Sadok, H., Reduced rank extrapolation applied to electronic structure computations, Electron. Trans. Numer. Anal., 38, 347-362, (2011) · Zbl 1287.65040 [12] El-Moallem, R.; Sadok, H., Vector extrapolation applied to algebraic Riccati equations arising in transport theory, Electron. Trans. Numer. Anal., 40, 489-506, (2013) · Zbl 1288.65057 [13] Ford, W. D.; Sidi, A., Recursive algorithms for vector extrapolation methods, Appl. Numer. Math., 4, 477-489, (1988) · Zbl 0649.65004 [14] Jbilou, K.; Sadok, H., Vector extrapolation methods. applications and numerical comparison, J. Comput. Appl. Math., 122, 149-165, (2000) · Zbl 0974.65034 [15] Napov, A.; Notay, Y., An algebraic multigrid method with guaranteed convergence rate, SIAM J. Sci. Comput., 34, A1079-A1109, (2012) · Zbl 1248.65037 [16] Y. Notay, AGMG software and documentation, available at http://homepages.ulb.ac.be/ ynotay/AGMG; Y. Notay, AGMG software and documentation, available at http://homepages.ulb.ac.be/ ynotay/AGMG [17] Notay, Y., An aggregation-based algebraic multigrid method, Electron. Trans. Numer. Anal., 37, 123-146, (2010) · Zbl 1206.65133 [18] Notay, Y., Aggregation-based algebraic multigrid for convection-diffusion equations, SIAM J. Sci. Comput., 34, A2288-A2316, (2012) · Zbl 1250.76139 [19] Raviart, P. A.; Thomas, J. M., A mixed finite element method for second order elliptic problems, (Galligani, L. I.; Magenes, E., Mathematical Aspects of the Finite Element Method, Lecture Notes in Math., # 606, (1977), Springer New York), 292-315 [20] Brezzi, F.; Douglas Jr., J.; Marini, L. D., Two families of mixed finite elements for second order elliptic problems, Numer. Math., 47, 217-235, (1985) · Zbl 0599.65072 [21] Brenner, S. C.; Scott, L. R., The mathematical theory of finite element methods, (2008), Springer New York · Zbl 1135.65042 [22] Brezzi, F.; Fortin, M., Mixed and hybrid finite element methods, (1991), Springer Berlin · Zbl 0788.73002 [23] Bahriawati, C.; Carstensen, C., Three MATLAB implementations of the lowest-order Raviart-Thomas MFEM with a posteriori control, Comput. Methods Appl. Math., 5, 333-361, (2005) · Zbl 1086.65107 [24] Brezzi, F.; Fortin, M.; Stenberg, R., Error analysis of mixed-interpolated elements for Reissner-Mindlin plates, Math. Models Methods Appl. Sci., 1, 125-151, (1991) · Zbl 0751.73053 [25] P.O. Persson, G. Strang, Simple mesh generator in MATLAB, available at http://persson.berkeley.edu/distmesh/; P.O. Persson, G. Strang, Simple mesh generator in MATLAB, available at http://persson.berkeley.edu/distmesh/ · Zbl 1061.65134 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.