Stability estimates and structural spectral properties of saddle point problems

For a general class of saddle point problems sharp estimates for Babuška’s inf-sup stability constants are derived in terms of the constants in Brezzi’s theory. In the finite-dimensional Hermitian case more detailed spectral properties of preconditioned saddle point matrices are presented, which are helpful for the convergence analysis of common Krylov subspace methods. The theoretical results are applied to two model problems from optimal control with time-periodic state equations. Numerical experiments with the preconditioned minimal residual method are reported.

[1]  Kent-André Mardal,et al.  Uniform preconditioners for the time dependent Stokes problem , 2006, Numerische Mathematik.

[2]  I. Babuska.  The finite element method with Lagrangian multipliers , 1973 .

[3]  D. Arnold.  Discretization by finite elements of a model parameter dependent problem , 1981 .

[4]  Ragnar Winther,et al.  A Preconditioned Iterative Method for Saddlepoint Problems , 1992, SIAM J. Matrix Anal. Appl..

[5]  Ulrich Langer,et al.  A Robust Preconditioned MinRes Solver for Time-periodic Eddy Current Problems , 2013, Comput. Methods Appl. Math..

[6]  Max D. Gunzburger,et al.  The Velocity Tracking Problem for Navier-Stokes Flows With Boundary Control , 2000, SIAM J. Control. Optim..

[7]  Maxim A. Olshanskii,et al.  On the Convergence of a Multigrid Method for Linear Reaction-Diffusion Problems , 2000, Computing.

[8]  Gundolf Haase,et al.  Adaptive Domain Decomposition Methods for Finite and Boundary Element Equations , 1997 .

[9]  J. Pasciak,et al.  Iterative techniques for time dependent Stokes problems , 1997 .

[10]  I. Babuska.  Error-bounds for finite element method , 1971 .

[11]  Frédéric Nataf,et al.  Spillane, N. and Dolean Maini, Victorita and Hauret, P. and Nataf, F. and Pechstein, C. and Scheichl, R. (2013) Abstract robust coarse spaces for systems of PDEs via generalized eigenproblems in the overlaps , 2018 .

[12]  Huidong Yang,et al.  Numerical simulation of fluid-structure interaction problems on hybrid meshes with algebraic multigrid methods , 2011, J. Comput. Appl. Math..

[13]  A. Wathen,et al.  Minimum residual methods for augmented systems , 1998 .

[14]  Kent-André Mardal,et al.  Uniform preconditioners for the time dependent Stokes problem , 2004, Numerische Mathematik.

[15]  Maxim A. Olshanskii,et al.  Uniform preconditioners for a parameter dependent saddle point problem with application to generalized Stokes interface equations , 2006, Numerische Mathematik.

[16]  W. Zulehner.  Non-standard Norms and Robust Estimates for Saddle Point Problems , 2011 .

[17]  Michel Fortin,et al.  Mixed and Hybrid Finite Element Methods , 2011, Springer Series in Computational Mathematics.

[18]  A. Berman.  CHAPTER 2 – NONNEGATIVE MATRICES , 1979 .

[19]  Christof Vömel,et al.  ScaLAPACK's MRRR algorithm , 2010, TOMS.

[20]  Valeria Simoncini,et al.  An Optimal Iterative Solver for Symmetric Indefinite Systems Stemming from Mixed Approximation , 2010, TOMS.

[21]  Nicholas I. M. Gould,et al.  Spectral Analysis of Saddle Point Matrices with Indefinite Leading Blocks , 2009, SIAM J. Matrix Anal. Appl..

[22]  F. Brezzi.  On the existence, uniqueness and approximation of saddle-point problems arising from lagrangian multipliers , 1974 .

[23]  Gene H. Golub,et al.  Numerical solution of saddle point problems , 2005, Acta Numerica.

[24]  Michael Kolmbauer,et al.  A preconditioned MinRes solver for time‐periodic parabolic optimal control problems , 2013, Numer. Linear Algebra Appl..

[25]  Robert J. Plemmons,et al.  Nonnegative Matrices in the Mathematical Sciences , 1979, Classics in Applied Mathematics.

[26]  Kent-André Mardal,et al.  Preconditioning discretizations of systems of partial differential equations , 2011, Numer. Linear Algebra Appl..

[27]  J. Lions.  Optimal Control of Systems Governed by Partial Differential Equations , 1971 .

[28]  M. Saunders,et al.  Solution of Sparse Indefinite Systems of Linear Equations , 1975 .

[29]  Joachim Schöberl,et al.  Symmetric Indefinite Preconditioners for Saddle Point Problems with Applications to PDE-Constrained Optimization Problems , 2007, SIAM J. Matrix Anal. Appl..

[30]  J. L. Lions,et al.  Control of Systems Governed by Parabolic Partial Differential Equations , 1971 .

[31]  Ralf Hiptmair,et al.  Operator Preconditioning , 2006, Comput. Math. Appl..

[32]  F. Tröltzsch.  Optimal Control of Partial Differential Equations: Theory, Methods and Applications , 2010 .

[33]  Anne Greenbaum,et al.  Iterative methods for solving linear systems , 1997, Frontiers in applied mathematics.

[34]  Stefan Ulbrich,et al.  Optimization with PDE Constraints , 2008, Mathematical modelling.

[35]  M. Kollmann,et al.  A Robust Preconditioner for Distributed Optimal Control for Stokes Flow with Control Constraints , 2013 .

[36]  Ludmil T. Zikatanov,et al.  Some observations on Babu\vs}ka and Brezzi theories , 2003, Numerische Mathematik.

[37]  Ulrich Langer,et al.  Domain Decomposition Solvers for Frequency-Domain Finite Element Equations , 2011 .

[38]  J. Cahouet,et al.  Some fast 3D finite element solvers for the generalized Stokes problem , 1988 .