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 H1({\Omega})-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(div)-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.
翻译:对涉及迁移、总压力和液体压力的Biot问题的三个场外变异配制,构建了事后误差估计。焦点的分解是 H1 ((@Omega}) 与 Taylor-Hood 相容的有限元素组合,包括用于迁移和流压总压力的多元度 k+ 1 和流压和 k。一个后误差估计器是根据H(div) 与压力和通量近似相容的重建结果得出的。重整压力的对称只能以微弱的方式得到满足。重整压力的数值结果可以在当地对一套顶端补补码进行,并导致对差错进行有保证的上限,常数仅取决于与补丁相关的本地常数,从而取决于三角的规律性。特别强调了几乎压材料,而且误差估计在可压缩的限度内保持统一。L型域的数值结果证实了理论,以及误差估计在适应性战略中的适当使用。