Biot 方程任意阶全混合有限元离散的分裂策略
摘要
我们研究 Biot 方程的全混合形式，其特征为流和位移之间的对称耦合。这种结构使我们能够为每个子物理问题使用稳定的混合有限元，而无需跨两个子物理问题强大的兼容性条件。为利用这种灵活性同时保持两个子问题的守恒结构，我们考虑其中对弹性应力张量对称性进行弱 enforcement 的全混合有限元方法。 resulting mixed formulation exhibits a saddle-point structure whose stability is determined by suitable inf--sup conditions. Inf--sup stability is established for several families of discrete spaces of arbitrary order, leading to optimal a priori error estimates. Iterative splitting strategies following the classical fixed-stress split with additional tuning are specifically investigated for the fully mixed formulation, with proof of convergence and rates depending on the coupling strength. Contrary to previous analyses on coupled problems with a symmetric structure, we theoretically prove the efficacy of negative stabilization, consistent with Schur-complement ideas. Numerical results based on analytical solutions and the classical Mandel problem support the theory.
关键词
引用
@article{arxiv.2603.18241,
title = {Splitting-strategies for arbitrary-order fully mixed finite element discretizations of the Biot equations},
author = {Fleurianne Bertrand and Jakub Wiktor Both and Tugay Dağlı},
journal= {arXiv preprint arXiv:2603.18241},
year = {2026}
}