In this work, we study the locking-free numerical method for a Barenblatt-Biot poroelastic model. When solving by the continuous Galerkin mixed finite element method, the model exists two kind of locking phenomena for special physical parameters. To overcome these locking phenomena, we introduce new variables to reformulate the original problem into a new problem, which exists a built-in mechanism to keep the continuous Galerkin mixed finite element method stable. It can be regarded as a generalized Stokes problem for two given and and two diffusion problems for a given . The generalized Stokes problem can be adopted by some stable solver and the diffusion problems can be solved by continuous Galerkin finite element. Moreover, the existence and uniqueness of weak solution is proved by using the standard Galerkin method and combining with priori estimates and some invariant quantities. After that, we design fully discrete time-stepping schemes to use mixed finite element method with element pairs for space variables and backward Euler method for time variable, and analyze the coupled and decoupled time stepping methods based on the proposed scheme. The optimal convergence order is obtained in both space and time. Finally, some numerical examples are presented to show the optimal convergence rates about variables and the robustness of proposed method with respect to , and to verify that there is no locking phenomenon.

中文翻译：

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Barenblatt-Biot 多孔弹性模型的新混合有限元方法分析


在这项工作中，我们研究了 Barenblatt-Biot 多孔弹性模型的无锁定数值方法。采用连续伽辽金混合有限元法求解时，模型存在两种针对特殊物理参数的锁定现象。为了克服这些锁定现象，我们引入新的变量将原始问题重新表述为新问题，该问题存在一种内置机制来保持连续伽辽金混合有限元方法的稳定。它可以被视为两个给定 和 的广义斯托克斯问题以及给定 的两个扩散问题。广义斯托克斯问题可以采用一些稳定的求解器，扩散问题可以用连续伽辽金有限元来求解。此外，利用标准伽辽金方法，结合先验估计和一些不变量，证明了弱解的存在唯一性。之后，我们设计了全离散时间步进方案，对空间变量采用元素对混合有限元法，对时间变量采用后向欧拉法，并对基于该方案的耦合和解耦时间步进方法进行了分析。在空间和时间上都得到了最优的收敛阶数。最后，给出了一些数值例子来展示变量的最优收敛速度以及所提出的方法对于 的鲁棒性，并验证不存在锁定现象。