Parabolic hysteresis problems revisited: Finite element error analysis and convergent Newton-type solvers

TL;DR

This paper provides rigorous finite element error analysis and develops a globally convergent damped smoothing Newton solver for parabolic PDEs with hysteresis, supported by numerical experiments confirming theoretical results.

Contribution

It introduces the first finite element error estimates for hysteresis-involving PDEs and proposes a robust Newton-type solver with proven convergence properties.

Findings

Numerical experiments confirm theoretical convergence rates.

The proposed solver outperforms existing methods in robustness and efficiency.

Higher-than-predicted convergence orders observed in quasilinear problems.

Abstract

Numerical investigations of partial differential equations with hysteresis have largely focused on simulations, leaving numerical error analysis unexplored and relying mainly on derivative-free nonlinear solvers. This work establishes rigorous finite element error estimates for the backward Euler fully discrete scheme applied to semilinear and quasilinear parabolic equations involving continuous hysteresis operators. To efficiently handle the inherent nonsmoothness of the resulting nonlinear algebraic systems, we develop a damped smoothing Newton solver under a general condition on the smoothing approximation, ensuring global convergence together with local Q-quadratic convergence. Numerical experiments confirm the theoretical convergence rates for semilinear problems, while showing higher-than-predicted orders for quasilinear ones. The robustness and efficiency of the proposed solver are further demonstrated in comparison with existing methods.