求解Brinkman方程的一种压力鲁棒协调间断有限元法  

A Pressure-robust Conforming Discontinuous Galerkin Finite Element Method for the Brinkman Equation

作　　者：陈俊杰 张莉 黄潇莹 CHEN Junjie;ZHANG Li;HUANG Xiaoying(School of Mathematical Science,Sichuan Normal University,Chengdu 610066,Sichuan;V.C.&V.R.Key Lab of Sichuan Province,Sichuan Normal University,Chengdu 610066,Sichuan)

机构地区：[1]四川师范大学数学科学学院,四川成都610066 [2]四川师范大学可视化计算与虚拟现实四川省重点实验室,四川成都610066

出　　处：《四川师范大学学报(自然科学版)》2025年第4期543-550,共8页Journal of Sichuan Normal University(Natural Science)

基　　金：国家自然科学基金(11871138);四川省科技计划项目(2022JDTD0019)。

摘　　要：针对Brinkman方程构建一种新的协调间断(conforming discontinuous Galerkin,CDG)有限元方法.该方法采用DG有限元空间,但其离散格式在形式上却保持协调有限元方法的简洁特征.使用H(div)空间逼近速度空间,借鉴弱有限元方法的思想,在变分格式中使用弱梯度算子▽_(w)代替传统意义的梯度算子▽,在▽_(w)v∈[Pk+1(T)]^(d×d)空间中近似▽_(v),构造出新的协调DG有限元方法具有如下优点:1)速度保持原方程无散度的物理性质;2)速度的误差与压力的误差无关,即压力鲁棒;3)不需要添加复杂的稳定项来增强切线方向速度的连续性.针对构建的离散格式,分析其稳定性和收敛性,并得到不同范数下速度和压力的最优误差估计,数值算例结果验证了理论结果.In this paper,a new conforming discontinuous Galerkin finite element method is proposed to solve the Brinkmanequation.This method adopts discontinuous finite element space,but its discrete format maintains the concisecharacteristics of the conforming finite element method in form.This method not only uses H(div)finite elementspace to approximate the velocity space,but also draws on the idea of the weak Galerkin finite element method,thatis,the weak gradient operator▽_(w)is used instead of the gradient operator in the variational form.Besides,the gradient operator ▽is approximated in▽_(w)v∈[P k+1(T)]^(d×d)space.The new conforming discontinuous Galerkin finite elementmethod has the following advantages:1)The velocity retains the physical property of the Brinkman equationwithout divergence.2)The error of the velocity is independent of the error of the pressure.3)There is no need toadd complex stabilization terms to enhance the continuity of the velocity in the tangential direction.For theconstructed discrete form,its stability and convergence are analyzed in detail,and velocity and pressure underdifferent norms are obtained.The theoretical results are verified by numerical examples.

关 键 词：Brinkman方程 压力鲁棒 H(div)空间 弱梯度算子 协调DG 

分 类 号：O211.6[理学—概率论与数理统计] O175.2[理学—数学] O211.4