We analyze the numerical performance of a preconditioning technique recently proposed in  for the efficient solution of parametrized linear systems arising from the finite element (FE) discretization of parameterdependent elliptic partial differential equations (PDEs). In order to exploit the parametric dependence of the PDE, the proposed preconditioner takes advantage of the reduced basis (RB) method within the preconditioned iterative solver employed to solve the linear system, and combines a RB solver, playing the role of coarse component, with a traditional fine grid (such as Additive Schwarz or block Jacobi) preconditioner. A sequence of RB spaces is required to handle the approximation of the error-residual equation at each step of the iterative method at hand, whence the name of Multi Space Reduced Basis (MSRB) method. In this paper, a numerical investigation of the proposed technique is carried on in the case of a Richardson iterative method, and then extended to the flexible GMRES method, in order to solve parameterized advection-diffusion problems. Particular attention is payed to the impact of anisotropic diffusion coefficients and (possibly dominant) transport terms on the proposed preconditioner, by carrying out detailed comparisons with the current state of the art algebraic multigrid preconditioners.
1. N. Dal Santo, S. Deparis, A. Manzoni, and A. Quarteroni, Multi space reduced basis preconditioners for large-scale parametrized PDEs, tech. rep., Mathicse 32, 2016.
2. S. Brenner and R. Scott, The mathematical theory of finite element methods, vol. 15. Springer Science & Business Media, 2007.
3. A. Ern and J.-L. Guermond, Theory and practice of finite elements. Number 159 in Applied Mathematical Sciences. Springer, New York, 2004.
4. A. Quarteroni, Numerical Models for Differential Problems, vol. 9 of Modeling, Simulation and Applications (MS&A). Springer-Verlag Italia, Milano, 2nd ed., 2014.
5. C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral methods in fluid dynamics. Springer-Verlag, 1988.
6. Y. Saad, Iterative methods for sparse linear systems. SIAM, 2003.
7. A. Quarteroni and A. Valli, Domain decomposition methods for partial differential equations, vol. 10. Oxford University Press, 1999.
8. A. J. Wathen, Preconditioning, Acta Numerica, vol. 24, pp. 329-376, 5 2015.
9. A. Toselli and O. B. Widlund, Domain decomposition methods: algorithms and theory. Springer series in computational mathematics, Berlin: Springer, 2005.
10. A. Quarteroni, A. Manzoni, and F. Negri, Reduced Basis Methods for Partial Differential Equations: An Introduction, vol. 92. Springer, 2016.
11. J. S. Hesthaven, G. Rozza, and B. Stamm, Certified reduced basis methods for parametrized partial differential equations, SpringerBriefs in Mathematics, 2016.
12. O. Zahm and A. Nouy, Interpolation of inverse operators for preconditioning parameterdependent equations, SIAM Journal on Scientific Computing, vol. 38, no. 2, pp. A1044- A1074, 2016.
13. K. Carlberg, V. Forstall, and R. Tuminaro, Krylov-subspace recycling via the podaugmented conjugate-gradient method, SIAM Journal on Matrix Analysis and Applications, vol. 37, no. 3, pp. 1304-1336, 2016.
14. Y. Saad, A flexible inner-outer preconditioned gmres algorithm, SIAM Journal on Scientific Computing, vol. 14, no. 2, pp. 461-469, 1993.
15. M. Bebendorf, Y. Maday, and B. Stamm, Comparison of some reduced representation approximations, in Reduced Order Methods for Modeling and Computational Reduction (A. Quarteroni and G. Rozza, eds.), vol. 9 of Modeling, Simulation and Applications, pp. 67-100, Cham, Switzerland: Springer, 2014.
16. A. Ramage, A multigrid preconditioner for stabilised discretisations of advection-diffusion problems, Journal of Computational and Applied Mathematics, vol. 110, no. 1, pp. 187 - 203, 1999.
17. L. Bertagna, S. Deparis, L. Formaggia, D. Forti, and A. Veneziani, The LifeV library: engineering mathematics beyond the proof of concept, arXiv:1710.06596, 2017.
18. M. W. Gee, C. M. Siefert, J. J. Hu, R. S. Tuminaro, and M. G. Sala, Ml 5.0 smoothed aggregation user’s guide, tech. rep., SAND2006-2649, Sandia National Laboratories, 2006.
19. N. Dal Santo, S. Deparis, A. Manzoni, and A. Quarteroni, An algebraic least squares reduced basis method for the solution of parametrized Stokes equations, tech. rep., Mathicse 21, 2017.
20. F. Ballarin, A. Manzoni, A. Quarteroni, and G. Rozza, Supremizer stabilization of POD- Galerkin approximation of parametrized steady incompressible Navier-Stokes equations, International Journal for Numerical Methods in Engineering, vol. 102, no. 5, pp. 1136-1161, 2015.