Sharp stability and approximation estimates for symmetric saddle point systems

References

• Bacuta C. Schur complements on Hilbert spaces and saddle point systems. J. Comput. Appl. Math. 2009;225:581–593.
   Web of Science ®Google Scholar
• Bacuta C. A unified approach for Uzawa algorithms. SIAM J. Numer. Anal. 2006;44:2633–2649.
   Web of Science ®Google Scholar
• Aziz A, Babuška I. Survey lectures on mathematical foundations of the finite element method. In: Aziz A, editor. The mathematical foundations of the finite element method with applications to partial differential equations (Proc. Sympos., Univ. Maryland, Baltimore, Md., 1972). New York (NY): Academic Press; 1972. p. 1–359.
   Google Scholar
• Babuška I. The finite element method with Lagrangian multipliers. Numer. Math. 1972/73;20:179–192.
   Web of Science ®Google Scholar
• Brezzi F. On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers. Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge. 1974;8:129–151.
   Google Scholar
• Braess D. Finite elements. Theory, fast solvers, and applications in solid mechanics. Cambridge: Cambridge University Press; 1997.
   Google Scholar
• Brenner S, Scott LR. The mathematical theory of finite element methods. New York (NY): Springer-Verlag; 1994.
   Google Scholar
• Falk RS, Osborn JE. Error estimates for mixed methods. RAIRO Anal. Numér. 1980;14:249–277.
   Google Scholar
• Arnold DN, Brezzi F, Fortin M. A stable finite element for the Stokes equations. Calcolo. 1985;21:337–344.
   Google Scholar
• Girault V, Raviart PA. Finite element methods for Navier–Stokes equations. Vol. 15. Berlin: Springer-Verlag; 1986.
   Google Scholar
• Brezzi F, Fortin M. Mixed and hybrid finite element methods. New York (NY): Springer-Verlag; 1991.
   Google Scholar
• Ern A, Guermond J-L. Theory and practice of finite elements. New York (NY): Springer-Verlag; 2004.
   Google Scholar
• Monk P. A mixed method for approximating Maxwell’s equations. SIAM J. Numer. Anal. 1991;28:1610–1634.
   Web of Science ®Google Scholar
• Quarteroni A, Valli A. Numerical approximation of partial differential equations. Berlin: Springer; 1994.
   Google Scholar
• Brezzi F. Stability of saddle-points in finite dimensions. In: Blowey J, Craig A, Shardlow T, editors. Frontiers in numerical analysis (Durham, 2002), Universitext. Berlin: Springer; 2003. p. 17–61.
   Google Scholar
• Elman HC, Silvester D, Wathen A. Finite elements and fast iterative solvers. Oxford: Oxford Science Publications; 2005.
   Google Scholar
• Mardal K-A, Winther R. Preconditioning discretizations of systems of partial differential equations. Numer. Linear Algebra Appl. 2011;18:1–40.
   Web of Science ®Google Scholar
• Kuznetsov YuA. Efficient iterative solvers for elliptic finite element problems on nonmatching grids. Russian J. Numer. Anal. Math. Model. 1995;10:187–211.
   Web of Science ®Google Scholar
• Silvester DJ, Wathen AJ. Fast & robust solvers for time-discretised incompressible Navier–Stokes equations. In: Numerical analysis 1995 (Dundee, 1995). Vol. 344, Pitman research notes in mathematics series. Harlow: Longman; 1996. p. 154–168.
   Google Scholar
• Murphy MF, Golub G, Wathen A. A note on preconditioning for indefinite linear systems. SIAM J. Sci. Comput. 2000;21:1969–1972 (electronic).
   Web of Science ®Google Scholar
• Benzi M, Golub G, Liesen J. Numerical solutions of saddle point problems. Acta Numerica. 2005;14:1–137.
   Google Scholar
• Schechter M. Principles of functional analysis. 2nd ed. Vol. 36, Graduate studies in mathematics. Providence (RI): American Mathematical Society; 2002.
   Google Scholar
• Xu J, Zikatanov L. Some observations on Babuška and Brezzi theories. Numer. Math. 2003;94:195–202.
   Web of Science ®Google Scholar
• Kato T. Estimation of iterated matrices, with application to the Von Neumann condition. Numer. Math. 1960;2:22–29.
   Google Scholar
• Sayas FJ. Infimum–supremum. Bol. Soc. Esp. Mat. Apl. Se⌃MA. 2007;41:19–40.
   Google Scholar
• Boffi D, Brezzi F, Demkowicz L, Durán RG, Falk R, Fortin M. Mixed finite elements, compatibility conditions, and applications. Vol. 1939, Lecture notes in mathematics. Berlin: Springer-Verlag; 2008. Fondazione C.I.M.E., Florence, Lectures given at the C.I.M.E. Summer School held in Cetraro, June 26-July 1, 2006, edited by Boffi and Lucia Gastaldi.
   Google Scholar
• Bacuta C. Subspace interpolation with applications to elliptic regularity. Numer. Funct. Anal. Optim. 2008;29:88–114.
   Web of Science ®Google Scholar
• Bacuta C, Bramble JH. Regularity estimates for solutions of the equations of linear elasticity in convex plane polygonal domains. Z. Angew. Math. Phys. (ZAMP). 2003;54:874–878.
   Web of Science ®Google Scholar
• Bacuta C, Bramble JH, Xu J. Regularity estimates for elliptic boundary value problems in Besov spaces. Math. Comp. 2003;72:1577–1595.
   Web of Science ®Google Scholar
• Bacuta C, Bramble JH, Xu J. Regularity estimates for elliptic boundary value problems with smooth data on polygonal domains. J. Numer. Math. 2003;11:75–94.
   Google Scholar
• Reed M, Simon B. Methods of modern mathematical physics. I. 2nd ed. Functional analysis. New York (NY): Academic Press [Harcourt Brace Jovanovich Publishers]; 1980.
   Google Scholar
• Bacuta C. Cascadic multilevel algorithms for symmetric saddle point systems. Comput. Math. Appl. 2014;67:1905–1913.
   Web of Science ®Google Scholar
• Bacuta C, Monk P. Multilevel discretization of symmetric saddle point systems without the discrete LBB condition. Appl. Numer. Math. 2012;62:667–681.
   Web of Science ®Google Scholar
• Bacuta C, Shu L. Multilevel gradient Uzawa algorithms for symmetric saddle point problems. J. Sci. Comput. 2013;57:105–123.
   Web of Science ®Google Scholar
• Bansch E, Morin P, Nocheto RH. An adaptive Uzawa fem for the Stokes problem: convergence without the inf-sup condition. SIAM J. Numer. Anal. 2002;40:1027–1229.
   Google Scholar