A posteriori error estimates are constructed for the three-field variational formulation of the Biot problem involving the displacements, the total pressure and the fluid pressure. The discretization under focus is the $H^1\left(\Omega \right)$-conforming Taylor–Hood finite element combination, consisting of polynomial degrees $k+1$ for the displacements and the fluid pressure and $k$ for the total pressure. An a posteriori error estimator is derived on the basis of $H\left(\text{div}\right)$-conforming reconstructions of the stress and flux approximations. The symmetry of the reconstructed stress is allowed to be satisfied only weakly. The reconstructions can be performed locally on a set of vertex patches and lead to a guaranteed upper bound for the error with a constant that depends only on local constants associated with the patches and thus on the shape regularity of the triangulation. Particular emphasis is given to nearly incompressible materials and the error estimates hold uniformly in the incompressible limit. Numerical results on the L-shaped domain confirm the theory and the suitable use of the error estimator in adaptive strategies.

后验误差估计是针对涉及位移，总压力和流体压力的比奥问题的三场变分公式而构建的。重点关注的离散化是$H^{\mathrm{1个}}（\Omega ）$多项式构成的符合标准的泰勒-胡德有限元组合 $ķ+\mathrm{1个}$ 排量和液压 $ķ$总压力。后验误差估计是基于$H（\text{div}）$应力和通量近似值的符合重构。重建应力的对称性只能微弱地满足。重构可以在一组顶点补丁上局部执行，并导致误差的保证上限，该常数的上限仅取决于与补丁相关联的局部常数，从而取决于三角剖分的形状规则性。特别强调几乎不可压缩的材料，并且误差估计均匀地保持在不可压缩的范围内。L形域上的数值结果证实了误差估计器的理论及其在自适应策略中的适当使用。