The Modified Weak Galerkin Finite Element Method for Solving Brinkman Equations

DOI：10.3770/j.issn:2095-2651.2019.06.011

 作者 单位 孙立娜 吉林大学数学学院, 吉林 长春 130012 封玥 吉林大学数学学院, 吉林 长春 130012 刘元元 北京应用物理与计算数学研究所, 北京 100083 张然 吉林大学数学学院, 吉林 长春 130012

本文针对Brinkman方程引入了一种修正弱Galerkin(MWG)有限元方法.我们通过具有两个离散弱梯度算子的变分形式来逼近模型. 在MWG方法中, 分别用次数为$k$和$k-1$的不连续分段多项式来近似速度函数$u$和压力函数$p$. MWG方法的主要思想是用内部函数的平均值代替边界函数. 因此, 与WG方法相比, MWG方法在不降低准确性的同时, 具有更少的自由度, 对于任意次数不超过$k-1$ 的多项式,MWG方法均可以满足稳定性条件. MWG 方法具有高度的灵活性, 它允许在具有一定形状正则性的任意多边形或多面体上使用不连续函数. 针对$H^1$和$L^22$范数下的速度和压力近似解, 建立了最优阶误差估计. 数值算例表明了该方法的准确性, 收敛性和稳定性.

A modified weak Galerkin (MWG) finite element method is introduced for the Brinkman equations in this paper. We approximate the model by the variational formulation based on two discrete weak gradient operators. In the MWG finite element method, discontinuous piecewise polynomials of degree $k$ and $k-1$ are used to approximate the velocity $\textbf{\textit{u}}$ and the pressure $p$, respectively. The main idea of the MWG finite element method is to replace the boundary functions by the average of the interior functions. Therefore, the MWG finite element method has fewer degrees of freedom than the WG finite element method without loss of accuracy. The MWG finite element method satisfies the stability conditions for any polynomial with degree no more than $k-1$. The MWG finite element method is highly flexible by allowing the use of discontinuous functions on arbitrary polygons or polyhedra with certain shape regularity. Optimal order error estimates are established for the velocity and pressure approximations in $H^1$ and $L^2$ norms. Some numerical examples are presented to demonstrate the accuracy, convergence and stability of the method.