发表机构
Peking University（北京大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对曲边域简支板问题中 Babuška 悖论导致的非预期极限，提出仅利用离散几何的边界修正混合有限元方法，将边界一致性误差从 h^(1/2) 提升至 h^(3/2)，并证明位移等量的最优收敛阶。
AI 中文摘要
经典的 Babuška 悖论表明，在曲边域的多边形近似上求解简支板问题，其解可能收敛到一个非预期的极限。我们针对简支 Kirchhoff-Love 板问题，发展了一种边界修正的 $H({\rm divdiv};\mathbb S)$-$L^2$ 混合有限元方法。该修正仅利用离散边界几何。将弯矩作为独立未知量引入，使得条件 $M_{nn}=0$ 可以直接施加，同时，对有效剪力的逐边均值约束抑制了主要的几何不一致性。该约束将边界一致性误差从 $\mathcal O(h^{1/2})$ 提升至 $\mathcal O(h^{3/2})$。相关的边界修正作用于 ${\rm divdiv}$ 的核中，保持离散平衡方程不变，从而得到一个一致稳定的格式。在适当的正则性假设下，我们证明了弯矩和后处理位移的间断 Hessian 的 $L^2$ 误差估计为 $h^{3/2}$ 阶，位移的误差估计为 $h^2$ 阶。该分析涵盖了多连通区域以及边界可能穿越物理边界的多边形近似。数值实验证实了这些收敛阶以及相对于未修正方法的改进。
英文摘要
The classical Babuška paradox shows that solutions of simply supported plate problems on polygonal approximations of a curved domain may converge to an unintended limit. We develop a boundary-corrected $H({\rm divdiv};\mathbb S)$-$L^2$ mixed finite element method for the simply supported Kirchhoff-Love plate problem. The correction uses only the discrete boundary geometry. Introducing the bending moment as an independent unknown allows the condition $M_{nn}=0$ to be imposed directly, while an edgewise mean constraint on the effective shear suppresses the leading geometric inconsistency. This constraint improves the boundary consistency error from $\mathcal O(h^{1/2})$ to $\mathcal O(h^{3/2})$. The associated boundary corrections act in the kernel of $\rm{divdiv}$, leaving the discrete equilibrium equation unchanged and yielding a uniformly stable scheme. Under suitable regularity assumptions, we prove $L^2$-error estimates of order $h^{3/2}$ for the bending moment and the broken Hessian of the postprocessed displacement, and of order $h^2$ for the displacement. The analysis covers multiply connected domains and polygonal approximations whose boundaries may cross the physical boundary. Numerical experiments confirm these rates and the improvement over the uncorrected method.
