Smoothing Newton Methods for Evolving Piecewise Smooth Systems Arising from Parabolic PDEs with Hysteresis

Abstract

Implicit discretizations of parabolic partial differential equations with hysteresis pose significant challenges for the design of robust nonlinear solvers due to the underlying history dependence. In this work, we derive a unified abstract formulation for the resulting nonlinear systems and show that their nonlinearities possess a monotone piecewise smooth structure with evolving switching behavior. Motivated by this structure, we develop a smoothing Newton method based on structure-preserving smoothing criteria, combining monotonicity preservation with an intermediate slope property. Under these conditions, the proposed method is shown to be globally convergent and locally Q-quadratically convergent. To enable practical implementation, we further introduce an explicit arc-based smoothing construction, thereby bridging the gap between abstract convergence theory and implementable algorithms. Numerical experiments for transient eddy-current problems with Preisach hysteresis demonstrate the robustness and efficiency of the proposed method compared with existing approaches.

arXiv comment33 pages. Revised and refocused version: finite-element error analysis removed; the analysis of the nonlinear structure and smoothing Newton solver expanded, with updated numerical experiments