This paper analyses conforming and nonconforming virtual element formulations of arbitrary polynomial degrees on general polygonal meshes for the coupling of solid and fluid phases in deformable porous plates. The governing equations consist of one fourth-order equation for the transverse displacement of the middle surface coupled with a second-order equation for the pressure head relative to the solid with mixed boundary conditions. We propose novel enrichment operators that connect nonconforming virtual element spaces of general degree to continuous Sobolev spaces. These operators satisfy additional orthogonal and best-approximation properties (referred to as a conforming companion operator in the context of finite element methods), which play an important role in the nonconforming methods. This paper proves a priori error estimates in the best-approximation form, and derives residual--based reliable and efficient a posteriori error estimates in appropriate norms, and shows that these error bounds are robust with respect to the main model parameters. The computational examples illustrate the numerical behaviour of the suggested virtual element discretisations and confirm the theoretical findings on different polygonal meshes with mixed boundary conditions.

翻译：本文分析了在一般多边形网格上，针对可变形多孔板中固相与液相耦合问题，具有任意多项式次数的协调与非协调虚拟单元格式。控制方程包括一个关于中面横向位移的四阶方程，以及一个关于相对于固体的压力水头的二阶方程，并附带混合边界条件。我们提出了新颖的富集算子，将一般次数的非协调虚拟单元空间与连续Sobolev空间相连接。这些算子满足额外的正交性和最佳逼近性质（在有限元方法背景下称为协调伴随算子），在非协调方法中发挥重要作用。本文证明了最佳逼近形式下的先验误差估计，推导了基于残差的可靠且高效的后验误差估计（在适当范数下），并表明这些误差界对主要模型参数具有鲁棒性。数值算例展示了所建议的虚拟单元离散格式的数值表现，并在不同多边形网格及混合边界条件下验证了理论结果。