A three-field Banach spaces-based mixed formulation for the unsteady Brinkman–Forchheimer equations
A three-field Banach spaces-based mixed formulation for the unsteady Brinkman–Forchheimer equations
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DOI：
10.1016/j.cma.2022.114895
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发表时间：
2022-05
影响因子：
7.2
通讯作者：
Sergio Caucao;Ricardo Oyarzúa;Segundo Villa-Fuentes;I. Yotov
中科院分区：
文献类型：
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作者：
Sergio Caucao;Ricardo Oyarzúa;Segundo Villa-Fuentes;I. Yotov
提出并分析了非定常流动Brinkman-Forchheimer方程的一种新的混合格式。除了速度，我们的方法还引入了速度梯度和伪应力张量作为进一步的未知数。作为结果，我们得到了基于三场Banach空间的混合变分公式，其中上述变量是系统的主要未知数。利用关于非线性单调算子的经典结果，我们建立了弱形式解的存在唯一性，并得到了相应的稳定界。然后，我们提出了一种基于伪应力张量的k次≥0元和速度和速度梯度的k次间断分段多项式的单纯网格上的半离散连续时间近似。此外，通过向后欧拉时间离散，我们引入了一种全离散的有限元格式。我们证明了两种格式的适定性，得到了两种格式的稳定界，并在网格上的准一致假设下，建立了相应的误差估计。我们给出了几个数值结果，验证了理论的收敛速度，并说明了该方法在一系列区域构型和模型参数下的性能和灵活性。
We propose and analyze a new mixed formulation for the Brinkman–Forchheimer equations for unsteady flows. Besides the velocity, our approach introduces the velocity gradient and a pseudostress tensor as further unknowns. As a consequence, we obtain a three-field Banach spaces-based mixed variational formulation, where the aforementioned variables are the main unknowns of the system. We establish existence and uniqueness of a solution to the weak formulation, and derive the corresponding stability bounds, employing classical results on nonlinear monotone operators. We then propose a semidiscrete continuous-in-time approximation on simplicial grids based on the Raviart–Thomas elements of degree k≥ 0 for the pseudostress tensor and discontinuous piecewise polynomials of degree k for the velocity and the velocity gradient. In addition, by means of the backward Euler time discretization, we introduce a fully discrete finite element scheme. We prove well-posedness and derive the stability bounds for both schemes, and under a quasi-uniformity assumption on the mesh, we establish the corresponding error estimates. We provide several numerical results verifying the theoretical rates of convergence and illustrating the performance and flexibility of the method for a range of domain configurations and model parameters.