A multipoint stress-flux mixed finite element method for the Stokes-Biot model
A multipoint stress-flux mixed finite element method for the Stokes-Biot model
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DOI：
10.1007/s00211-022-01310-2
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发表时间：
2020-11
影响因子：
2.1
通讯作者：
Sergio Caucao;Tongtong Li;I. Yotov
中科院分区：
文献类型：
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作者：
Sergio Caucao;Tongtong Li;I. Yotov
在本文中，我们提出并分析了一个完全混合的自由流体和多孔弹性介质之间的相互作用所产生的耦合问题的提法。在自由流体和多孔弹性区域的流动分别由Stokes和毕奥方程控制，并且传输条件由质量守恒、应力平衡和Beavers-Joseph-Saffman定律给出。 我们在这两个域中应用双混合配方，其中的斯托克斯和多孔弹性应力张量的对称性是通过设置涡度和结构旋转张量作为辅助未知数。反过来，由于传输条件变得必不可少，他们被强加弱通过引入的痕迹的流体速度，结构速度，和孔隙弹性介质压力的接口作为相关的拉格朗日乘子。建立了一个连续的弱配方，以及一个半离散的连续时间与非匹配网格的配方，连同相应的稳定性界限的解决方案的存在性和唯一性。此外，我们开发了一个新的多点应力通量混合有限元方法涉及顶点求积规则，它允许局部消除的应力，旋转，和达西通量。通过数值实验对全离散格式的适定性和误差分析以及相应的收敛速度进行了验证。
In this paper we present and analyze a fully-mixed formulation for the coupled problem arising in the interaction between a free fluid and a poroelastic medium. The flows in the free fluid and poroelastic regions are governed by the Stokes and Biot equations, respectively, and the transmission conditions are given by mass conservation, balance of stresses, and the Beavers-Joseph-Saffman law. We apply dual-mixed formulations in both domains, where the symmetry of the Stokes and poroelastic stress tensors is imposed by setting the vorticity and structure rotation tensors as auxiliary unknowns. In turn, since the transmission conditions become essential, they are imposed weakly by introducing the traces of the fluid velocity, structure velocity, and the poroelastic media pressure on the interface as the associated Lagrange multipliers. The existence and uniqueness of a solution are established for the continuous weak formulation, as well as a semidiscrete continuous-in-time formulation with non-matching grids, together with the corresponding stability bounds. In addition, we develop a new multipoint stress-flux mixed finite element method by involving the vertex quadrature rule, which allows for local elimination of the stresses, rotations, and Darcy fluxes. Well-posedness and error analysis with corresponding rates of convergence for the fully-discrete scheme are complemented by several numerical experiments.