论文标题
高度异质多孔介质中自适应模型的适应性和变异数值方案
Well-posedness and variational numerical scheme for an adaptive model in highly heterogeneous porous media
论文作者
论文摘要
多孔介质中流体流的数学模型通常由连续性方程和选择的本构定律描述。后者取决于手头的问题，可能是流体压力梯度和速度之间的非线性关系。这种关系的实际形状通常是在问题开始时选择的，尽管实际上，流体可能会在其适用性范围内经历速度。我们在这里提出了一个自适应模型，以便根据计算的速度在本地选择最合适的法律。从分析的角度来看，当法律在速度中单调并在一个空间维度中显示出存在时，我们表现出问题的良好性。从计算的角度来看，我们提出了一种新方法，基于通过模拟正规化的基础耗散，即流体通过阻力损失到多孔介质的功率。结果表明，在单调案例中，使用$γ$ - convergence会收敛到原始问题。这种方法产生了一种适用于非常普遍的问题的变异数值方案，并且我们在三个测试用例上验证。
Mathematical modeling of fluid flow in a porous medium is usually described by a continuity equation and a chosen constitutive law. The latter, depending on the problem at hand, may be a nonlinear relation between the fluid's pressure gradient and velocity. The actual shape of this relation is normally chosen at the outset of the problem, even though, in practice, the fluid may experience velocities outside of its range of applicability. We propose here an adaptive model, so that the most appropriate law is locally selected depending on the computed velocity. From the analytical point of view, we show well-posedness of the problem when the law is monotone in velocity and show existence in one space dimension otherwise. From the computational point of view, we present a new approach based on regularizing via mollification the underlying dissipation, i.e., the power lost by the fluid to the porous medium through drag. The resulting regularization is shown to converge to the original problem using $Γ$-convergence on the dissipation in the monotone case. This approach gives rise to a variational numerical scheme which applies to very general problems and which we validate on three test cases.
