Efficient Solvers for Large-Scale Saddle Point Systems Arising in Feedback Stabilization of Multi-field Flow Problems

This article introduces a block preconditioner to solve large-scale block structured saddle point systems using a Krylov-based method. Such saddle point systems arise, e.g., in the Riccati-based feedback stabilization approach for multi-field flow problems as discussed in [2]. Combining well known approximation methods like a least-squares commutator approach for the Navier-Stokes Schur complement, an algebraic multigrid method, and a Chebyshev-Semi-Iteration, an efficient preconditioner is derived and tested for different parameter sets by using a simplified reactor model that describes the spread concentration of a reactive species forced by an incompressible velocity field.

[1]  Yvan Notay,et al.  Aggregation-Based Algebraic Multigrid for Convection-Diffusion Equations , 2012, SIAM J. Sci. Comput..

[2]  K. Stüben.  Algebraic multigrid (AMG): experiences and comparisons , 1983 .

[3]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[4]  Peter Benner,et al.  Efficient Solution of Large-Scale Saddle Point Systems Arising in Riccati-Based Boundary Feedback Stabilization of Incompressible Stokes Flow , 2013, SIAM J. Sci. Comput..

[5]  P. Benner,et al.  Optimal Control-Based Feedback Stabilization of Multi-field Flow Problems , 2014 .

[6]  Danny C. Sorensen,et al.  Balanced Truncation Model Reduction for a Class of Descriptor Systems with Application to the Oseen Equations , 2008, SIAM J. Sci. Comput..

[7]  J. Oden,et al.  Finite Element Methods for Flow Problems , 2003 .

[8]  M. Benzi,et al.  INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS Int. J. Numer. Meth. Fluids (2010) Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/fld.2267 Modified augmented Lagrangian preconditioners for the incompressible Navier , 2022 .

[9]  Peter Benner,et al.  Riccati-based boundary feedback stabilization of incompressible Navier-Stokes flow , 2013 .

[10]  Jean-Pierre Raymond,et al.  HINFINITY Feedback Boundary Stabilization of the Two-Dimensional Navier-Stokes Equations , 2011, SIAM J. Control. Optim..

[11]  Y. Saad,et al.  GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems , 1986 .

[12]  Howard C. Elman,et al.  Finite Elements and Fast Iterative Solvers: with Applications in Incompressible Fluid Dynamics , 2014 .

[13]  Martin Stoll,et al.  All-at-once solution of time-dependent Stokes control , 2013, J. Comput. Phys..

[14]  Yousef Saad,et al.  A Flexible Inner-Outer Preconditioned GMRES Algorithm , 1993, SIAM J. Sci. Comput..

[15]  A. Spence,et al.  Eigenvalues of Block Matrices Arising from Problems in Fluid Mechanics , 1994, SIAM J. Matrix Anal. Appl..

[16]  Y. Notay.  An aggregation-based algebraic multigrid method , 2010 .

[17]  John N. Shadid,et al.  Block Preconditioners Based on Approximate Commutators , 2005, SIAM J. Sci. Comput..

[18]  Andreas Griewank,et al.  Trends in PDE Constrained Optimization , 2014 .

[19]  Artem Napov,et al.  An Algebraic Multigrid Method with Guaranteed Convergence Rate , 2012, SIAM J. Sci. Comput..

[20]  E. Wagner.  International Series of Numerical Mathematics , 1963 .

[21]  Barry Lee,et al.  Finite elements and fast iterative solvers: with applications in incompressible fluid dynamics , 2006, Math. Comput..

[22]  P TaylorC.Hood,et al.  Navier-Stokes equations using mixed interpolation , 1974 .

[23]  Jean-Pierre Raymond,et al.  Feedback Boundary Stabilization of the Two-Dimensional Navier--Stokes Equations , 2006, SIAM J. Control. Optim..

[24]  J. W. Ruge,et al.  4. Algebraic Multigrid , 1987 .