Hellan-Herrmann-Johnson 方法的局部最大范数误差估计
Localized maximum-norm error estimates for the Hellan-Herrmann-Johnson method
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中文总结 AI 辅助
针对夹紧 Kirchhoff 板问题，提出 HHJ 方法弯矩的局部最大范数误差估计，通过保持核的局部化技术实现最优收敛，并揭示奇偶次多项式下的对称恢复增益。
中文摘要 AI 辅助
我们推导了夹紧 Kirchhoff 板问题中由 Hellan-Herrmann-Johnson 方法计算的弯矩的局部最大范数界。主要困难在于在不离开 HHJ 空间或违反其核约束的情况下局部化离散应力。利用对称旋度势，我们构造了一个保持核的局部化，并将局部 Green 应力与全局离散 Green 应力联系起来。这一论证将局部插值误差与较弱的全局污染项分离开来。在对辅助问题没有显式正则性假设的情况下，所得估计给出了弯矩值和单元一阶导数的最优逐点收敛性。对于正多项式次数，没有对数损失，而最低阶值估计保留了对数因子。在额外的反射对称性下，对称恢复改善了偶数次多项式的弯矩值和奇数次多项式的一阶导数。证明的增益分别为阶的三分之一和二分之一。数值实验证实了这种奇偶依赖性，并显示出接近一个完整阶的增益，超过了理论上建立的增益。
英文摘要
We derive localized maximum-norm bounds for bending moments computed by the Hellan-Herrmann-Johnson method for the clamped Kirchhoff plate problem. The main difficulty is to localize the discrete stress without leaving the HHJ space or violating its kernel constraint. Using symmetric-curl potentials, we construct a kernel-preserving localization and connect local Green stresses with a global discrete Green stress. This argument separates the local interpolation error from a weaker global pollution term. Under explicit regularity assumptions on the auxiliary problems, the resulting estimates give optimal pointwise convergence for bending-moment values and elementwise first derivatives. No logarithmic loss occurs for positive polynomial degrees, whereas the lowest-order value estimate retains a logarithmic factor. Under additional reflection symmetry, symmetric recovery improves bending-moment values for even polynomial degrees and first derivatives for odd degrees. The proved gains are one third and one half of an order, respectively. Numerical experiments confirm this parity dependence and exhibit gains close to one full order, exceeding those established theoretically.
发表机构
- School of Mathematical Sciences, Zhejiang University（浙江大学数学科学学院）
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