Darcy-Forchheimer方程的新型稳定化混合有限元格式

The derivation of new mixed finite element methods for Darcy-Forchheimer model and related problems has received considerable attention in the past twenty years. A tremendous amount of work has been done on the numerical solution of Darcy-Forchheimer model based on the primal mixed formulation and the dual mixed formulation. Despite the extensive efforts in this area, research on the finite element formulations is still at its infancy. In this project, several stabilized mixed finite element methods based on the primal mixed formulation and the dual mixed formulation will be developed and analyzed for Darcy-Forchheimer model. The velocities and the pressures will be approximated by the lowest-equal-order pair of conforming and nonconforming finite elements. To enhance the stability properties of the methods, a symmetric, weakly consistent stabilization operator will be added to the standard Galerkin formulation. It is worthwhile to emphasize that the stabilizing term does not couple to all the terms of the residual and is therefore independent of both time derivatives, source terms and higher derivatives. Another important feature is that the stabilization parameter is independent of the diffusion coefficient and more generally has less dependence on the problem parameters than in the classical stabilization methods. To linearize the mixed-method equations, we use a two-grid algorithm based on the fixed point iteration method. By using the two-grid method, solving a nonlinear equation on a fine grid is reduced to solving a nonlinear equation on a coarse grid together with solving a linear equation on a fine grid. As a result, solving such a large class of non-linear equation will not be much more difficult than solving one single linearized equation. The stability and convergence of these new methods will be analyzed. Then, numerical experiments will be carried out to confirm our theoretical results. Our work will extend the application range of finite elements, and provide a powerful supplement to existing numerical methods.

Darcy-Forchheimer模型广泛应用于多孔介质和裂隙介质中流体流动问题的数值模拟，其混合有限元法的构造和分析一直是计算数学领域研究的热点问题。近年来，基于primal 混合变分形式和dual 混合变分形式的混合有限元格式受到越来越多的关注。但是，关于其稳定化有限元法的研究尚未展开。本项目的主要目的就是提出并分析几类Darcy-Forchheimer模型基于primal混合变分形式和dual混合变分形式的对称稳定化有限元法，对速度和压力采用最低等阶有限元对逼近，并通过添加基于非残差的对称稳定项来强化格式的稳定性。其次，为了高效求解对应的离散方程组，我们拟构造建立在不动点迭代法基础上的混合有限元两重网格算法。最后，我们将分析这些方法的稳定性与收敛性，并通过数值算例验证我们的理论结果。本项目的工作可以拓宽稳定化有限元法的应用范围，是对已有工作有力的补充和完善。

Darcy-Forchheimer方程刻画了多孔介质中单相流体在高速流动时速度与压力梯度之间的非线性关系，具有非线性和单调性特征，关于其混合有限元方法的理论研究一直是一个热点问题。为了获得稳定的有限元解，经典的混合有限元理论要求速度和压力有限元空间必须满足所谓的inf-sup条件。然而，工程上计算方便的等阶有限元空间却并不满足inf-sup条件。克服这一困难的有效途径之一是引入稳定化方法强化混合有限元法的稳定性。本项目的主要工作正是构造并分析Darcy-Forchheimer方程及相关问题的新型稳定化混合有限元法。首先，我们研究了Darcy-Forchheimer方程的节点投影稳定化混合有限元方法。该方法是一类基于局部Scott-Zhang投影算子的对称稳定化方法，采用等阶有限元空间逼近速度和压力，通过在标准的有限元格式上添加额外的具有对称性质的速度散度和压力梯度波动控制项强化格式的稳定性，绕开了inf-sup条件束缚。稳定项的运算仅在局部单元上进行，不用进行高阶导数或者边界积分的运算，易于编程实现。我们证明了有限元解的存在唯一性，给出了速度和压力有限元解的拟最优阶L2-模误差估计。据我们所知，目前还没有类似的研究报道。其次，我们将节点投影稳定化混合有限元方法推广应用到pseudostress-速度形式的Stokes方程，采用等阶有限元空间来逼近非对称的pseudostress和速度，通过添加额外的具有对称性质的pseudostress散度和速度梯度波动控制项强化格式的稳定性，证明了pseudostress和速度有限元解的最优阶L2-模误差估计，这一结果极大地改进了已有的结果。最后，我们针对退化形式的Darcy-Forchheimer方程——具有Dirichlet边界条件的Darcy方程，提出了修正的块中心有限差分方法。这一方法可以视为在适当的求积公式下的最低阶Raviart-Thomas混合有限元方法，速度和压力的离散解具有二阶超收敛特性，改进了著名计算数学家Arbogast，Wheeler和Yotov的工作。