粗网格或畸变网格的后验误差估计
A posteriori error estimation for coarse or distorted meshes
- Universidad de la República（乌拉圭共和国大学）
- Université de Strasbourg（斯特拉斯堡大学）
- Inria（法国国家数字研究院）
- Universidad Técnica Federico Santa María（费德里科·圣玛丽亚技术大学）
- Univ. Lille（里尔大学）
- Arts et Métiers Institute of Technology（巴黎高科艺术工艺学院）
- Centrale Lille（里尔中央理工学院）
- Junia（朱尼亚应用科学联盟）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本研究探讨平衡通量估计器在粗网格或畸变网格下的后验误差估计能力，其推导无需形状正则性等条件，仅受网格质量影响局部效率，不影响可靠性。
AI中文摘要：
针对真实几何（如患者特定解剖结构、扫描工业部件、断裂介质）生成的网格开展有限元模拟时，几乎无法满足经典后验误差分析所依赖的形状正则性假设。因此，从业者无法回答一个合理问题：当网格较粗且包含形状极差的单元时，模拟结果的误差有多大？本研究探讨平衡通量估计器在多大程度上可回答该问题。平衡通量估计器能给出能量范数离散误差的保证且完全可计算的上界，其常数为1；关键在于，该估计器的推导未使用形状正则性、拟均匀性或角度条件，仅依赖单个单元上的Poincaré常数，Payne和Weinberger证明该常数在任意凸单元K上最大为hK/π（hK为单元尺寸）。网格质量影响局部效率（即估计器的精度），但从不影响其可靠性，因此网格畸变时的失效模式是估计结果偏保守，而非错误置信。
英文摘要：
Finite element simulations carried out on meshes issued from real-world geometries - patient-specific anatomies, scanned industrial parts, fractured media - almost never satisfy the shape-regularity assumptions under which classical a posteriori error analysis is written. Practitioners are therefore left without a quantitative answer to a very legitimate question: how wrong is my simulation when my mesh is coarse and contains badly shaped elements? In this work, we study to which extent equilibrated flux estimators can provide such an answer. Indeed, they deliver a guaranteed and fully computable upper bound of the energy-norm discretization error, with a constant equal to one; and, crucially for the present study, its derivation uses no shape-regularity, quasi-uniformity or angle condition. The only geometric ingredient is the Poincaré constant on a single cell, which Payne and Weinberger showed to be at most hK /pi on any convex cell K, however distorted (hK being the cell size). Mesh quality affects the local efficiency - the sharpness of the estimator - but never its reliability, so that the failure mode under distortion is pessimism rather than false confidence.
