We present and analyze a robust and mass conservative virtual element method for the three-field formulation of Biot’s consolidation problem in poroelasticity. The displacement and fluid flux are respectively approximated by enriched \({\varvec{H}}({\text {div}})\) virtual elements and \({\varvec{H}}({\text {div}})\) virtual elements, while the pressure is discretized by piecewise polynomial functions. Optimal a priori error estimates are obtained, including the semi-discrete scheme and the fully-discrete scheme with the implicit Euler approximation in time. Moreover, our method achieves robustness with respect to the constants hidden in the error estimates, even for the Lamé coefficient tending to infinity and the arbitrarily small constrained specific storage coefficient, and therefore it is free of both volumetric (Poisson) locking and nonphysical pressure oscillations. Meanwhile, it also conserves pointwise mass conservation for Biot’s consolidation problem on the discrete level.

我们提出并分析了一种稳健且质量保守的虚拟元方法，用于求解 Biot 在孔隙弹性中的固结问题的三场公式。位移和流体通量分别由丰富的\({\varvec{H}}({\text {div}})\)虚拟元素和\({\varvec{H}}({\text {div}} )\)虚拟元素，而压力通过分段多项式函数离散化。得到了最优的先验误差估计，包括半离散方案和时间上具有隐式欧拉近似的全离散方案。此外，我们的方法对隐藏在误差估计中的常数实现了鲁棒性，即使对于趋于无穷大的拉梅系数和任意小的约束比存储系数，它也没有体积（泊松）锁定和非物理压力振荡. 同时，它还在离散水平上对 Biot 固结问题守恒点质量守恒。