arXiv:0907.2600v2 [math.NA] 20 Jan 2010 Multigrid and preconditioning strategies for implicit PDE solvers for degenerate parabolic equations M. Donatelli * , M. Semplice † and S. Serra-Capizzano ‡ Dipartimento di Fisica e Matematica Universit` a dell’Insubria Via Valleggio 11, 22100 Como, Italy. October 30, 2018 Abstract The novel contribution of this paper relies in the proposal of a fully im- plicit numerical method designed for nonlinear degenerate parabolic equa- tions, in its convergence/stability analysis, and in the study of the related computational cost. In fact, due to the nonlinear nature of the underly- ing mathematical model, the use of a fixed point scheme is required and every step implies the solution of large, locally structured, linear systems. A special effort is devoted to the spectral analysis of the relevant matrices and to the design of appropriate iterative or multi-iterative solvers, with special attention to preconditioned Krylov methods and to multigrid pro- cedures: in particular we investigate the mutual benefit of combining in various ways suitable preconditioners with V-cycle algorithms. Numerical experiments in one and two spatial dimensions for the validation of our multi-facet analysis complement this contribution. AMS SC: 65N12, 65F10 (65N22, 15A18, 47B35) 1 Introduction We consider a single equation of the form ∂u ∂t = ∇· (D(u)∇u) , (1) where D(u) is a non-negative function. The the equation is parabolic and it called degenerate whenever D(u) vanishes for some values of u. For the conver- gence analysis of our numerical methods, we will require that D(u) is at least * Email: marco.donatelli@uninsubria.it † Corresponding author. Email: matteo.semplice@uninsubria.it ‡ Email: stefano.serrac@uninsubria.it 1