AI 中文总结
本文提出非参数化非协调基于面的有限元方法，适用于复杂多面体网格上的椭圆问题，避免参数映射导致的逼近退化，并保证稳定收敛与最优收敛速率。
AI 中文摘要
在这项工作中，我们提出并分析了用于复杂混合三维网格（由任意多面体单元组成）上椭圆问题的低阶非协调基于面的有限元方法。重点在于标量问题，但该方法可以以张量方式扩展，以设计混合公式的经典Crouzeix--Raviart和Rannacher--Turek方法的推广。我们采用局部逼近空间的非参数化构造。我们证明，这避免了在扭曲网格上由参数化映射通常引起的逼近性质退化。利用这些逼近空间，我们为非常一般的单元定义了有限元格式。我们进一步证明，在严重退化的网格上，受间断伽辽金方法启发的途径能够消除一致性误差，即使在存在潜在弯曲面的情况下也是如此。给出了棱柱（三角形基底）和金字塔（四边形基底）单元的多项式空间构造的显式示例。所得到的格式在理论上和数值上都被证明是稳定且收敛的，具有最优收敛速率。
英文摘要
In this work, we propose and analyze low-order nonconforming face-based finite element methods for elliptic problems on complex hybrid three-dimensional meshes, composed of arbitrary polytopal cells. The focus is on scalar problems, but the approach can be extended in a tensor manner to design generalizations of the classical Crouzeix--Raviart and Rannacher--Turek methods for mixed formulations. We adopt a nonparametric construction of the local approximation spaces. We demonstrate that it avoids the degradation of approximation properties commonly induced by parametric mappings on distorted meshes. Using these approximation spaces, we define finite element schemes for very general cells. We further demonstrate that, on strongly degraded meshes, an approach inspired by discontinuous Galerkin methods enables removing consistency errors, even in the presence of potentially curved faces. Explicit examples of the construction of the polynomial space are given for prismatic (of triangular basis) and pyramidal (of quadrilateral basis) cells. The resulting schemes are shown theoretically and numerically to be stable and convergent, with an optimal convergence rate.
Journal refESAIM: Mathematical Modelling and Numerical Analysis, 2026, 60 (5), pp.2489-2515
