论文标题
Darcy-Forchheimer模型的多点通量混合有限元法的广义多尺度近似
Generalized multiscale approximation of a multipoint flux mixed finite element method for Darcy-Forchheimer model
论文作者
论文摘要
在本文中，我们为高度异质的多孔介质中的Darcy-Forchheimer模型提出了一种多尺度方法。该问题是在通用多尺度有限元方法（GMSFEM）与多点通量混合有限元（MFMFE）方法的框架中解决的。 ％在MFMFE方法中，采用适当的混合有限元元素空间和合适的正交规则，可以消除局部速度并导致以细胞为中心的压力系统。我们考虑使用最低顺序Brezzi-Douglas-Marini（$ \ textrm {bdm} _1 $）的MFMFE方法，用于速度和压力近似。对称梯形正交规则用于与速度变量相关的双线性形式的整合，以便允许消除局部速度，并导致以细胞为中心的压力系统。由简单组成的网格和$ h^2 $的平行四边形。我们为压力构建多尺度空间，并在GMSFEM框架之后解决粗网格上的问题。在离线阶段，我们构建本地快照空间并执行光谱分解，以使尺寸较小的离线空间获得较小的尺寸。在在线阶段，我们使用牛顿迭代算法来解决非线性问题并获得离线解决方案，从而减少了与标准PICARD迭代相比的迭代时间。基于离线空间和离线解决方案，我们计算在线基础功能，这些功能包含重要的全局信息，以迭代地迭代多尺度空间。在线基础功能是有效且准确的，可以大大减少相对错误。提供了数值示例，以突出提出的多尺度方法的性能。
In this paper, we propose a multiscale method for the Darcy-Forchheimer model in highly heterogeneous porous media. The problem is solved in the framework of generalized multiscale finite element methods (GMsFEM) combined with a multipoint flux mixed finite element (MFMFE) method. %In the MFMFE methods, appropriate mixed finite element spaces and suitable quadrature rules are employed, which allow for local velocity elimination and lead to a cell-centered system for the pressure. We consider the MFMFE method that utilizes the lowest order Brezzi-Douglas-Marini ($\textrm{BDM}_1$) mixed finite element spaces for the velocity and pressure approximation. The symmetric trapezoidal quadrature rule is employed for the integration of bilinear forms relating to the velocity variables so that the local velocity elimination is allowed and leads to a cell-centered system for the pressure. %on meshes composed of simplices and $h^2$-perturbed parallelograms. We construct multiscale space for the pressure and solve the problem on the coarse grid following the GMsFEM framework. In the offline stage, we construct local snapshot spaces and perform spectral decompositions to get the offline space with a smaller dimension. In the online stage, we use the Newton iterative algorithm to solve the nonlinear problem and obtain the offline solution, which reduces the iteration times greatly comparing to the standard Picard iteration. Based on the offline space and offline solution, we calculate online basis functions which contain important global information to enrich the multiscale space iteratively. The online basis functions are efficient and accurate to reduce relative errors substantially. Numerical examples are provided to highlight the performance of the proposed multiscale method.
