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arXiv · 2609.23676

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

Shu Xu·Liqun Cao
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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.

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Shu Xu, Liqun Cao. 2026-09-20. Numerical analysis of parabolic equations with Prandtl--Ishlinskii hysteresis of play type. https://arxiv.org/abs/2609.23676

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