AI 中文总结
研究比奥固结模型，基于$H(\rm div \, \rm div)$协调有限元提出一族混合-混合有限元方法，能逐点保持角动量及流体质量平衡，证明了离散问题适定性并建立最优先验误差估计。
AI 中文摘要
我们考虑比奥准静态固结模型的四场公式，将（有效）弹性应力、固体位移、流体通量和流体压力作为感兴趣的未知物理量。由动量和质量平衡方程组成的系统的弱形式，辅以应力-应变关系和达西定律，产生了一个双重摄动鞍点问题。在自然选择的无限维希尔伯特空间中证明了其适定性，其中位移-通量对在一个新颖的希尔伯特空间中求解。接下来，基于最近关于$H(\rm div \, \rm div)$协调有限元的工作，我们提出一族混合-混合有限元方法，该方法逐点保持角动量以及流体质量平衡，即从强意义上保持。我们证明了所得离散问题的适定性，并为相关的完全守恒混合-混合有限元方法族建立了最优先验误差估计。
英文摘要
We consider a four-field formulation of Biot's quasi-static model of consolidation with the (effective) elastic stress, the solid displacement, the fluid flux and the fluid pressure as unknown physical quantities of interest. The weak form of this system composed of the momentum and mass balance equations complemented by the stress-strain relation and Darcy's law yields a two-fold perturbed saddle-point problem. Its well-posedness is proven for a natural choice of inifinite-dimensional Hilbert spaces, where the displacement-flux pair is sought in a novel Hilbert space. Next, based on recent works on $H(\rm div \, \rm div)$-conforming finite elements, we propose a family of mixed-mixed finite element methods that preserve the angular momentum as well as the fluid mass balance pointwise, i.e., in a strong sense. We prove the well-posedness of the arising discrete problem and establish optimal a priori error estimates for the related family of fully conservative mixed-mixed finite element methods.
