多点通量混合有限元解耦算法研究

Solving saddle-type algebraic systems derived from the mixed finite element method on polygonal meshes is a challenging task, which restricts the use of efficient iterative solvers..Fortunately, the multipoint flux mixed finite element method preserves the advantages of the mixed finite element method and decouples the saddle-point problem. The method is based on the lowest order Brezzi-Douglas-Marini mixed element space, a special quadrature rule is employed that allows for local velocity elimination and leads to a cell-centered system for the pressures. This project intends to study the efficient decoupling algorithm of the porous media flow model, specifically: 1) the multipoint flux mixed finite element-characteristic finite element methods are used to solve Darcy miscible displacement problems, highly decoupled and fast and stable numerical schemes are constructed, theoretical deduction of the optimal convergence order and numerical experiments to verify the effectiveness of the algorithms are also presented. 2) Non-Darcy flow multipoint flux mixed element method are also considered, design robust iterative scheme and the decoupling algorithm, and present the regularity and the L^{p} (p≥2) error estimate for solutions. 3) Combining adaptive finite element and characteristic line technique, the decoupling algorithms for Non-Darcy miscible displacement problems are discussed. 4) Aiming at the optimal design problem of water flooding, optimal control problems of two-phase incompatible miscible displacement model are proposed to realize the fast and effective simulation of the more non-homogeneous nonlinear reservoir problem, hoping to serve the research and development of efficient simulation software in oil extraction.

多边形网格上混合有限元方法导出的鞍点型代数系统求解是一项具有挑战性的工作，其限制高效迭代求解器的使用。而多点通量混合有限元方法保持混合元优点且解耦鞍点问题。该方法基于最低阶Brezzi-Douglas-Marini混合元空间，采用特殊求积公式局部消除速度，解耦方程为压力块中心差分格式。本项目开展渗流问题高效解耦算法研究，具体地：1）研究达西混溶驱动问题多点通量混合元-特征有限元方法，构造高度解耦快速稳定数值格式，理论推导收敛阶并设计实验验证算法有效性；2）研究非达西流多点通量混合元方法，设计稳健迭代格式与解耦算法，推导解的正则性及L^{p}(p≥2)估计；3）结合自适应有限元和特征线技术，研究非达西混溶驱动问题解耦算法；4）针对油田注水采油优化设计问题，研究受控于两相不可压缩混溶驱动模型的最优控制问题，实现对真实非均质非线性油藏快速有效的模拟，以期服务于石油开采中高效模拟软件的研究和开发。

多边形网格上混合有限元方法导出的鞍点型代数系统求解是一项具有挑战性的工作，其限制高效迭代求解器的使用。而多点通量混合有限元方法保持混合元优点且解耦鞍点问题。该方法基于最低阶Brezzi-Douglas-Marini混合元空间，采用特殊求积公式局部消除速度，解耦方程为压力块中心差分格式。本项目主要开展渗流问题高效解耦算法研究，包括可压缩达西和不可压非达西缩混溶驱动问题多点通量混合有限元方法、Brinkman方程的稳定混合有限元方法以及不可压缩Darcy混溶驱动问题守恒特征有限元--多点通量混合有限元解耦算法研究，针对混溶驱动问题构造了高度解耦快速稳定数值格式，理论推导了收敛阶，得到了速度、压力和浓度的最优收敛阶，并设计实验验证算法有效性实现对真实非均质非线性油藏快速有效的模拟，以期服务于石油开采中高效模拟软件的研究和开发。