Multigrid and preconditioning strategies for implicit PDE solvers for degenerate parabolic equations
arXiv:0907.2600
Abstract
The novel contribution of this paper relies in the proposal of a fully implicit numerical method designed for nonlinear degenerate parabolic equations, in its convergence/stability analysis, and in the study of the related computational cost. In fact, due to the nonlinear nature of the underlying 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 procedures: 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.
29 pages, 10 figures. This version contains the linear algebra results of v1, with corrections and extensions. The applications (second part of v1) will be published soon in a separate paper