Multigrid and preconditioning strategies for implicit PDE solvers for degenerate parabolic equations

TL;DR

This paper introduces a fully implicit numerical method for nonlinear degenerate parabolic equations, focusing on convergence, stability, computational cost, and advanced solver strategies including multigrid and preconditioning techniques.

Contribution

It presents a new implicit scheme for degenerate parabolic equations, with detailed spectral analysis and innovative solver combinations, validated through numerical experiments.

Findings

Effective multigrid and preconditioned Krylov methods improve solver efficiency.

Spectral analysis guides the design of tailored iterative solvers.

Numerical validation confirms the method's stability and accuracy.

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.