Brinkman-Forchheimer方程的加罚有限元方法

Brinkman-Forchheimer方程(BF方程)是具有强非线性项并满足无散度条件的流动控制方程,其中无散度条件的精确满足对控制方程的数值求解极其重要.为了放松无散度条件的限制,本文采用了加罚方法.为了得到加罚问题解的适定性,首先,利用加罚关系将压力项消去,证明了速度所满足的具有单调性的非线性椭圆变分问题等价于对应能量泛函的极小化问题,从而得到了速度的存在唯一性.进一步,利用LBB条件证明了BF方程加罚问题压力的存在唯一性.其次,证明了BF方程加罚问题的Galerkin变分问题的解关于加罚参数收敛到BF方程的Galerkin变分问题的解.最后,给出了BF方程加罚问题Galerkin变分问题的有限维逼近问题及其解的存在唯一性,并且得出了采用协调有限元离散的误差估计.数值算例表明加罚方法是有效的.

Penalty Finite Element Approximation for the Brinkman-Forchheimer Equations

Brinkman-Forchheimer equations(BF equations)describe the motion of the incom-pressible fluid under the strong nonlinearities. The accurate treatment of the incompressibility condition is critical for the numerical treatment of the BF equations. The penalty treatment is introduced to relax the incompressibility condition. In order to obtain the well-posedness of the penalty problem, the pressure term is eliminated by using the penalty term, and an equivalence between the monotonous nonlinear elliptical problem and a minimization prob-lem of corresponding energy functional is proposed. From the LBB condition, the existence and uniqueness of the variational problem are obtained. The convergence with respected to the penalty parameter is proved. Finally, the existence and uniqueness of the finite dimen-sional approximating problem are derived, and the error estimate based on the conforming finite element discretization is obtained. Numerical results show that the penalty finite element approximation is effective.