AI 中文总结
本文针对Prandtl--Ishlinskii play型滞回抛物方程，提出隐式Euler $P_1$有限元离散，证明$O(h+\ au)$误差界，无需额外正则性，适用于空间Hilbert空间上的广义模型。
AI 中文摘要
对于具有滞回的抛物型方程的数值逼近的严格误差分析仍然有限，即使对于广泛使用的Prandtl--Ishlinskii play型滞回也是如此。在这项工作中，我们为隐式Euler $P_1$有限元离散化建立了$O(h+\ au)$误差界。该分析既不需要滞回变量的更高阶时间正则性（在滞回演化中通常无法期望），也不需要这些变量的额外空间正则性。对于时间离散化，我们利用加权Hilbert空间中的凸次梯度流结构以及相关的耗散和强制次梯度余项来获得一阶收敛。对于空间离散化，仅将扩散场限制在有限元空间中，并且保约束比较产生$O(h)$半离散估计。该分析是针对直接在空间Hilbert空间上表述的play型Prandtl--Ishlinskii算子开发的，涵盖了典型的逐点模型以及更一般的空间结构化约束。
英文摘要
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.
