AI 中文总结
本文开发了弱对称且无迹的切向-法向张量有限元，应用于 Brinkman 方程，构造了无稳定化、压力鲁棒的混合方法，并证明了最优误差估计和参数一致边界层估计。
AI 中文摘要
我们开发了一族在任意空间维度和所有多项式阶数下弱对称且逐点无迹的切向-法向张量有限元。对称性通过局部单元矩施加，而唯一全局耦合的应力自由度为切向-法向面矩；不需要顶点自由度。在三维及更高维度中，一个最低阶线性富集恢复了离散 Korn 稳定性所需的刚体运动面控制。作为主要应用，我们使用物理粘性应力为不可压缩 Brinkman 方程构造了一个分布混合方法。与散度相容的 BDM 速度和间断压力耦合，该方法无需稳定化，关于粘性参数一致稳定，精确无散，且压力鲁棒。我们在自然范数中建立了最优阶误差估计。在适当的参数显式正则性假设下，我们还获得了具有最优 Darcy 逼近阶的参数一致边界层估计。放宽切向-法向连续性产生代数等价的应力混合公式和无稳定化的虚拟元实现。
英文摘要
We develop a family of weakly symmetric and pointwise traceless tangential-normal tensor finite elements in arbitrary space dimension and for all polynomial orders. Symmetry is imposed through local cell moments, while the only globally coupled stress degrees of freedom are tangential-normal facet moments; no vertex degrees of freedom are required. In dimensions three and higher, a lowest-order linear enrichment restores the rigid-motion facet control required for discrete Korn stability. As a principal application, we construct a distributional mixed method for the incompressible Brinkman equations using the physical viscous stress. Coupled with divergence-conforming BDM velocities and discontinuous pressures, the method is stabilization-free, uniformly stable with respect to the viscosity parameter, exactly divergence-free, and pressure-robust. We establish optimal-order error estimates in the natural norms. Under suitable parameter-explicit regularity assumptions, we also obtain a parameter-uniform boundary-layer estimate with optimal Darcy approximation order. Relaxing tangential-normal continuity yields an algebraically equivalent stress-hybridized formulation and a stabilization-free virtual element realization.
Comments26 pages, 2 figures, 2 tables
