论文标题
Darcy的多孔媒体定律，具有多个微观结构
Darcy's Law for Porous Media with Multiple Microstructures
论文作者
论文摘要
在本文中，我们研究了带有多个微观结构的穿孔域中dirichlet问题的均匀化。首先，在子域之间的界面是Lipschitz表面的结合的情况下，我们表明有效的速度和压力受Darcy定律的控制，在该法律中，渗透率矩阵是分段常数的。关键步骤是使用牙垢的测试功能方法证明有效压力在整个界面之间是连续的。其次，假设界面满足某些平滑度和几何条件，我们建立了速度和压力收敛性的急剧误差估计。这是通过构建两个校正器来实现的。其中一个用于纠正界面上两尺度近似的不连续性，而另一个用于纠正解决方案边界值及其近似之间的差异。
In this paper we study the homogenization of the Dirichlet problem for the Stokes equations in a perforated domain with multiple microstructures. First, under the assumption that the interface between subdomains is a union of Lipschitz surfaces, we show that the effective velocity and pressure are governed by a Darcy law, where the permeability matrix is piecewise constant. The key step is to prove that the effective pressure is continuous across the interface, using Tartar's method of test functions. Secondly, we establish the sharp error estimates for the convergence of the velocity and pressure, assuming the interface satisfies certain smoothness and geometric conditions. This is achieved by constructing two correctors. One of them is used to correct the discontinuity of the two-scale approximation on the interface, while the other is used to correct the discrepancy between boundary values of the solution and its approximation.
