We study and compare five different combinations of finite element spaces for approximating the coupled flow and solid deformation system, so-called Biot’s equations. The permeability and porosity fields are heterogeneous and depend on solid displacement and fluid pressure. We provide detailed comparisons among the continuous Galerkin, discontinuous Galerkin, enriched Galerkin, and two types of mixed finite element methods. Several advantages and disadvantages for each of the above techniques are investigated by comparing local mass conservation properties, the accuracy of the flux approximation, number of degrees of freedom (DOF), and wall and CPU times. Three-field formulation methods with fluid velocity as an additional primary variable generally require a larger number of DOF, longer wall and CPU times, and a greater number of iterations in the linear solver in order to converge. The two-field formulation, a combination of continuous and enriched Galerkin function space, requires the fewest DOF among the methods that conserve local mass. Moreover, our results illustrate that three out of the five methods conserve local mass and produce similar flux approximations when conductivity alteration is included. These comparisons of the key performance indicators of different combinations of finite element methods can be utilized to choose the preferred method based on the required accuracy and the available computational resources.

我们研究和比较了有限元空间的五种不同组合，以近似耦合的流动和固体变形系统，即所谓的比奥方程。渗透率和孔隙率场是不均匀的，并取决于固体位移和流体压力。我们提供了连续Galerkin，不连续Galerkin，富集Galerkin和两种混合有限元方法之间的详细比较。通过比较局部质量守恒特性，通量近似值的准确性，自由度（DOF）数以及墙和CPU时间，研究了上述每种技术的几个优点和缺点。以流体速度为附加主要变量的三场公式化方法通常需要更多的自由度，更长的壁和CPU时间，并在线性求解器中进行大量迭代以收敛。两场公式是连续和丰富的Galerkin函数空间的组合，在节省局部质量的方法中需要最少的自由度。此外，我们的结果表明，当包括电导率变化时，五种方法中的三种可以节省局部质量并产生相似的通量近似值。可以将有限元方法不同组合的关键性能指标的这些比较用于根据所需的精度和可用的计算资源来选择首选方法。我们的结果表明，当包括电导率变化时，五种方法中的三种可以节省局部质量并产生相似的通量近似值。可以将有限元方法不同组合的关键性能指标的这些比较用于根据所需的精度和可用的计算资源来选择首选方法。我们的结果表明，当包括电导率变化时，五种方法中的三种可以节省局部质量并产生相似的通量近似值。可以将有限元方法不同组合的关键性能指标的这些比较用于根据所需的精度和可用的计算资源来选择首选方法。