具有曲界面表面流与多孔介质流耦合问题的有限元法

项目名称： 具有曲界面表面流与多孔介质流耦合问题的有限元法

项目编号： No.11301267

项目类型： 青年科学基金项目

立项/批准年度： 2014

项目学科： 数理科学和化学

项目作者： 黄佩奇

作者单位： 南京林业大学

项目金额： 22万元

中文摘要： 表面流与多孔介质流耦合问题由于在水文学、环境科学和生物流体动力学等领域有着广泛应用已成为人们研究的热点。由于实际问题中表面流与多孔介质流的交界面是弯曲的，国内外关于具有曲界面耦合问题有限元法的研究较少。本项目拟针对Stokes-Darcy这一典型模型，首次研究子问题在曲界面上耦合时的有限元方法。首先，考虑耦合问题的全局离散网格跟曲界面非匹配时的浸入界面有限元方法；接着，给出两个子问题的离散网格在曲界面上非匹配时的Mortar型有限元方法；然后，对上述有限元离散系统设计高效求解器。重点验证离散问题对应鞍点问题的LBB条件并证明有限元解的误差估计，得到求解器收敛的一致性。

中文关键词： 浸入界面有限元；Mortar方法；Nitsche方法；解耦；非匹配界面罚

英文摘要： The coupling problem of surface flow and porous media flow arouses increasing interest because of its significance in hydrology, environmental science and bio-fluid dynamics. Because the interface between surface flow and porous media flow is actually curve, and there is few studies about finite element method for this coupled problem with curved interface. We study firstly the finite element methods for the typical Stokes-Darcy model when the subproblems couple on the curved interface. First, we discuss the immersed interface finite element method for the coupling problem when the global discrete grids are unfitted with the curved interface. Then, a mortar-type finite element method is proposed when meshes in different subdomains are allowed to be nonmatching on the common curved interface. Moreover, we present some efficient solvers for the linear system discreted by these finite elements. We will try to derive the LBB condition for the saddle point problems carried out by the discrete forms and prove error estimates of the proposed finite element method. Furthermore, the uniform convergence rates of the solvers will also be given.

英文关键词： Immersed interface finite element；Mortar method；Nitsche method；Decoupling；Unfitted interface penalty