We develop a mixed finite element domain decomposition method on non-matching grids for the Biot system of poroelasticity. A displacement–pressure vector mortar function is introduced on the interfaces and utilized as a Lagrange multiplier to impose weakly continuity of normal stress and normal velocity. The mortar space can be on a coarse scale, resulting in a multiscale approximation. We establish existence, uniqueness, stability, and error estimates for the semi-discrete continuous-in-time formulation under a suitable condition on the richness of the mortar space. We further consider a fully-discrete method based on the backward Euler time discretization and show that the solution of the algebraic system at each time step can be reduced to solving a positive definite interface problem for the composite mortar variable. A multiscale stress–flux basis is constructed, which makes the number of subdomain solves independent of the number of iterations required for the interface problem, and weakly dependent on the number of time steps. We present numerical experiments verifying the theoretical results and illustrating the multiscale capabilities of the method for a heterogeneous benchmark problem.

中文翻译：

---

多孔弹性 Biot 系统的多尺度砂浆混合有限元方法


我们为多孔弹性的 Biot 系统在非匹配网格上开发了一种混合有限元域分解方法。在界面上引入了位移-压力矢量砂浆函数，并用作拉格朗日乘子，以施加法向应力和法向速度的弱连续性。砂浆空间可以是粗略的，从而产生多尺度近似。我们在砂浆空间的丰富度的适当条件下建立了半离散连续时间公式的存在性、唯一性、稳定性和误差估计。我们进一步考虑了一种基于后向欧拉时间离散化的全离散方法，并表明代数系统在每个时间步的解可以简化为求解复合砂浆变量的正定界面问题。构建了一个多尺度应力-通量基，这使得子域求解的数量与界面问题所需的迭代次数无关，并且弱依赖于时间步长的数量。我们提出了验证理论结果的数值实验，并说明了该方法对异构基准问题的多尺度能力。