论文标题
BIOT问题的弱对称应力重建估计后验错误估计
A posteriori error estimates by weakly symmetric stress reconstruction for the Biot problem
论文作者
论文摘要
为涉及位移，总压力和流体压力的BIOT问题的三场变异公式构建了后验误差估计。焦点下的离散化是H1（ω）合并的泰勒 - 霍德元素组合，由位移的多项式K + 1组成，总压力的流体压力和k组成。根据H（div）对应力和磁通近似的重构进行了后验误差估计器。仅允许轻弱地满足重建的应力的对称性。可以在一组顶点贴片上在本地执行重建，并导致误差的上限，其常数仅取决于与斑块相关的局部常数，从而取决于三角调节的形状规则性。特别强调几乎不可压缩的材料，并且误差估计值均匀地保持在不可压缩的极限下。 L形域上的数值结果证实了自适应策略中误差估计器的合适使用。
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(Ω)-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.
