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

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.