HSL_MI20: an efficient AMG preconditioner for finite element problems in 3D. Algebraic multigrid (AMG) is one of the most effective iterative methods for the solution of large, sparse linear systems obtained from the discretization of second-order scalar elliptic self-adjoint partial differential equations. It can also be used as a preconditioner for Krylov subspace methods. In this communication, we report on the design and development of a robust, effective and portable Fortran 95 implementation of the classical Ruge-St”uben AMG, which is available as package HSL_MI20 within the HSL mathematical software library. The routine can be used as a `black-box’ preconditioner, but it also offers the user a range of options and parameters. Proper tuning of these parameters for a particular application can significantly enhance the performance of an AMG-preconditioned Krylov solver. This is illustrated using a number of examples arising in the unstructured finite element discretization of the diffusion, the convection-diffusion, and the Stokes equations, as well as transient thermal convection problems associated with the Boussinesq approximation of the Navier-Stokes equations in 3D.

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  1. Cao, Shan-Mou; Wang, Zeng-Qi: PMHSS iteration method and preconditioners for Stokes control PDE-constrained optimization problems (2021)
  2. Benner, Peter; Onwunta, Akwum; Stoll, Martin: A low-rank inexact Newton-Krylov method for stochastic eigenvalue problems (2019)
  3. Bootland, Niall; Bentley, Alistair; Kees, Christopher; Wathen, Andrew: Preconditioners for two-phase incompressible Navier-Stokes flow (2019)
  4. Porcelli, Margherita; Simoncini, Valeria; Stoll, Martin: Preconditioning PDE-constrained optimization with (L^1)-sparsity and control constraints (2017)
  5. Robbe, Pieterjan; Nuyens, Dirk; Vandewalle, Stefan: A multi-index quasi-Monte Carlo algorithm for lognormal diffusion problems (2017)
  6. Benner, Peter; Dolgov, Sergey; Onwunta, Akwum; Stoll, Martin: Low-rank solvers for unsteady Stokes-Brinkman optimal control problem with random data (2016)
  7. Benner, Peter; Onwunta, Akwum; Stoll, Martin: Block-diagonal preconditioning for optimal control problems constrained by PDEs with uncertain inputs (2016)
  8. McDonald, Eleanor; Wathen, Andy: A simple proposal for parallel computation over time of an evolutionary process with implicit time stepping (2016)
  9. Palitta, Davide; Simoncini, Valeria: Matrix-equation-based strategies for convection-diffusion equations (2016)
  10. Pearson, John W.: Fast iterative solvers for large matrix systems arising from time-dependent Stokes control problems (2016)
  11. Wu, Yirong; Wang, Heyu: An AMG preconditioner for solving the Navier-Stokes equations with a moving mesh finite element method (2016)
  12. Bertolazzi, Enrico; Frego, Marco: Preconditioning complex symmetric linear systems (2015)
  13. Pearson, John W.: On the development of parameter-robust preconditioners and commutator arguments for solving Stokes control problems (2015)
  14. Porcelli, Margherita; Simoncini, Valeria; Tani, Mattia: Preconditioning of active-set Newton methods for PDE-constrained optimal control problems (2015)
  15. Wang, Zeng-Qi: Restrictively preconditioned Chebyshev method for solving systems of linear equations (2015)
  16. Ahmed, Sarfraz; Goodyer, Christopher E.; Jimack, Peter K.: An adaptive finite element procedure for fully-coupled point contact elastohydrodynamic lubrication problems (2014)
  17. Pestana, J.; Wathen, A. J.: Combination preconditioning of saddle point systems for positive definiteness. (2013)
  18. Ramage, Alison; Gartland, Eugene C. jun.: A preconditioned nullspace method for liquid crystal director modeling (2013)
  19. Sesana, Debora; Simoncini, Valeria: Spectral analysis of inexact constraint preconditioning for symmetric saddle point matrices (2013)
  20. Traverso, L.; Phillips, T. N.; Yang, Y.: Mixed finite element methods for groundwater flow in heterogeneous aquifers (2013)

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