Numerical analysis of parabolic equations with Prandtl--Ishlinskii hysteresis of play type

Abstract

Rigorous error analysis for numerical approximations of parabolic equations with hysteresis remains limited, even for the widely used Prandtl--Ishlinskii hysteresis of play type. In this work, we establish an $O(h+τ)$ error bound for an implicit Euler $P_1$ finite element discretization. The analysis requires neither higher-order temporal regularity of the hysteresis variables, which cannot in general be expected in hysteretic evolutions, nor additional spatial regularity of these variables. For the temporal discretization, we exploit a convex subgradient-flow structure in a weighted Hilbert space together with the associated dissipation and coercive subgradient remainder to obtain first-order convergence. For the spatial discretization, only the diffusive field is restricted to the finite element space, and a constraint-preserving comparison yields an $O(h)$ semidiscrete estimate. The analysis is developed for play-type Prandtl--Ishlinskii operators formulated directly on a spatial Hilbert space, encompassing the canonical pointwise model as well as more general spatially structured constraints.