Stokes-Darcy耦合问题的非匹配Nitsche方法

Because the interface of multiphysics coupled model is actually curve, and there is no any numerical work about Stokes-Darcy coupled problem with curved interface at home and abroad. We study firstly the discretized methods and their efficient solvers for this typical multidomain coupled problem using the unfitted Nitsche method presented recently, when the discrete grids of the coupled problem are unfitted with the curve interface. We will finish the following four questions: (1) use unfitted Nitsche method to solve Stokes interface problem; (2) devise unfitted Nitsche method to solve Stokes-Darcy coupled problem by using the same mixed element for the subproblems; (3) design unfitted Nitsche method to solve Stokes-Darcy coupled problem using different stable mixed elements for subproblems; (4) construct iterative algorithms for the discretized systems of these problems. We will prove that the convergence of the unfitted Nitsche method is independent of the position of the interface relative to the discrete mesh; divise the optimal preconditioners by analysing the condition number of the discrete system; present numerical experiments to confirm the theoretical results.

由于实际问题中多物理场耦合模型的交界面是弯曲的，而国内外没有关于具曲界面Stokes-Darcy耦合问题数值方法的研究。因此我们针对此多区域耦合问题，首次运用最近提出的非匹配Nitsche方法研究耦合问题的离散网格跟曲界面非匹配时的离散方法及其离散系统的高效求解器。具体解决如下四个问题：(1)非匹配Nitsche方法求解Stokes界面问题；(2)对子问题用同一个混合元离散，构造非匹配Nitsche方法求解Stokes-Darcy耦合问题；(3)对子问题用不同的稳定的混合元离散，构造非匹配Nitsche方法求解Stokes-Darcy耦合问题；(4)对上述三个问题的离散系统构造迭代算法。我们将证明非匹配Nitsche方法的收敛性与界面在离散网格中的位置无关；分析离散系统的条件数，构造最优的预条件子；给出数值算例验证理论结果。

Stokes-Darcy耦合问题由于其在水文学、环境科学和生物流体动力学等领域有着广泛应用已经成为人们研究的热点问题。而实际问题中多物理场耦合模型的交界面一般都是弯曲的，我们首次针对Stokes-Darcy这一典型多区域耦合问题，运用最近提出的非匹配Nitsche方法研究耦合问题的离散网格跟曲界面非匹配时的离散方法。具体解决了如下三个问题：(1)非匹配Nitsche方法求解Stokes界面问题；(2)对子问题用同一个混合元离散，构造非匹配Nitsche方法求解Stokes-Darcy耦合问题；(3)对子问题用不同的稳定的混合元离散，构造非匹配Nitsche方法求解Stokes-Darcy耦合问题。我们重点分析了离散格式对应鞍点问题的LBB条件；证明了非匹配Nitsche方法的稳定性和收敛性与界面在离散网格中的位置无关；给出数值算例验证理论结果。