This paper addresses the construction and analysis of a class of domain decomposition methods for the iterative solution of the quasi-static Biot problem in three-field formulation. The considered discrete model arises from time discretization by the implicit Euler method and space discretization by a family of strongly mass-conserving methods exploiting $H^{div}$-conforming approximations of the solid displacement and fluid flux fields. For the resulting saddle-point problem, we construct monolithic overlapping domain decomposition (DD) methods whose analysis relies on a transformation into an equivalent symmetric positive definite system and on stable decompositions of the involved finite element spaces under proper problem-dependent norms. Numerical results on two-dimensional test problems are in accordance with the provided theoretical uniform convergence estimates for the two-level multiplicative Schwarz method.

翻译：本文针对三场公式中拟静态Biot问题的迭代求解，研究了一类区域分解方法的构造与分析。所考虑的离散模型源于隐式欧拉方法的时间离散化，以及利用固体位移与流体通量场的$H^{div}$相容逼近实现的一族强质量守恒方法在空间上的离散化。针对所得到的鞍点问题，我们构建了一体化重叠型区域分解（DD）方法，其分析依赖于将问题转化为等价的对称正定系统，并在恰当的问题相关范数下对涉及到的有限元空间进行稳定分解。二维测试问题的数值结果与两层乘法Schwarz方法所提供的理论一致收敛性估计相吻合。