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 model of play type. In this work, we establish an O(h+τ) error bound for an implicit Euler P1 finite element discretization. The analysis requires neither higher-order temporal regularity nor additional spatial regularity of the hysteresis variables. For the temporal discretization, we exploit a convex subgradient-flow structure in a weighted Hilbert space and estimates of residual monotone terms 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 extension as well as more general spatially structured constraints.
Related
- Asymptotic Stability of Steady Prandtl Equations with Suction
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BibTeX
@misc{xu2026numerical,
title = {{Numerical analysis of parabolic equations with Prandtl--Ishlinskii hysteresis of play type}},
author = {Shu Xu and Liqun Cao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.23676},
eprint = {2609.23676},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23676},
}
RIS
TY - UNPB
TI - Numerical analysis of parabolic equations with Prandtl--Ishlinskii hysteresis of play type
AU - Shu Xu
AU - Liqun Cao
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2609.23676
UR - https://arxiv.org/abs/2609.23676
AB - Rigorous error analysis for numerical approximations of parabolic equations with hysteresis remains limited, even for the widely used Prandtl--Ishlinskii model 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 nor additional spatial regularity of the hysteresis variables. For the temporal discretization, we exploit a convex subgradient-flow structure in a weighted Hilbert space and estimates of residual monotone terms 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 extension as well as more general spatially structured constraints.
ER -