Pseudo-time regularization for PDE with solution-dependent diffusion

TL;DR

This paper introduces a unified regularized pseudo-time framework for solving nonmonotone PDEs with solution-dependent diffusion, combining convergence analysis, adaptive algorithms, and acceleration strategies, demonstrated on challenging diffusion problems.

Contribution

It presents a novel regularization approach that stabilizes Newton-like iterations for complex PDEs, with new analysis and adaptive strategies for improved convergence.

Findings

Effective stabilization of Newton-like iterations for nonmonotone PDEs.

Demonstrated convergence and acceleration on anisotropic diffusion problems.

Validated the method on problems with steep gradients and thin diffusion layers.

Abstract

This work unifies pseudo-time and inexact regularization techniques for nonmonotone classes of partial differential equations, into a regularized pseudo-time framework. Convergence of the residual at the predicted rate is investigated through the idea of controlling the linearization error, and regularization parameters are defined following this analysis, then assembled in an adaptive algorithm. The main innovations of this paper include the introduction of a Picard-like regularization term scaled by its cancellation effect on the linearization error to stabilize the Newton-like iteration; an updated analysis of the regularization parameters in terms of minimizing an appropriate quantity; and, strategies to accelerate the algorithm into the asymptotic regime. Numerical experiments demonstrate the method on an anisotropic diffusion problem where the Jacobian is not continuously differentiable, and a model problem with steep gradients and a thin diffusion layer.