In this paper, we consider flow and transport problems in thin domains. Modeling problems in thin domains occur in many applications, where thin domains lead to some type of reduced models. A typical example is one dimensional reduced-order model for flows in pipe-like geometries (e.g., blood vessels). In many reduced-order models, the model equations are described apriori by some analytical approaches. In this paper, we propose the use of multiscale methods, which are alternative to reduced-order models and can represent reduced-dimension modeling by using fewer basis functions (e.g., the use of one basis function corresponds to one dimensional approximation).

The mathematical model considered in the paper is described by a system of equations for velocity, pressure, and concentration, where the flow is described by the Stokes equations, and the transport is described by an unsteady convection-diffusion equation with non-homogeneous boundary conditions on walls (reactive boundaries). We start with the finite element approximation of the problem on unstructured grids and use it as a reference solution for two and three-dimensional model problems. Fine grid approximation resolves complex geometries on the grid level and leads to a large discrete system of equations that is computationally expensive to solve. To reduce the size of the discrete systems, we develop a multiscale model reduction technique, where we construct local multiscale basis functions to generate a lower-dimensional model on a coarse grid. The proposed multiscale model reduction is based on the Discontinuous Galerkin Generalized Multiscale Finite Element Method (DG-GMsGEM). In DG-GMsFEM for flow problems, we start with constructing the snapshot space for each interface between coarse grid cells to capture possible flows. For the reduction of the snapshot space size, we perform a dimension reduction via a solution of the local spectral problem and use eigenvectors corresponding to the smallest eigenvalues as multiscale basis functions for the approximation on the coarse grid. For the transport problem, we construct multiscale basis functions for each interface between coarse grid cells and present additional basis functions to capture non-homogeneous boundary conditions on walls. Finally, we will present some numerical simulations for three test geometries for two and three-dimensional problems to demonstrate the method's performance.

在本文中，我们考虑薄域中的流动和传输问题。许多应用程序中都会出现薄域中的建模问题，其中薄域会导致某种类型的简化模型。一个典型的例子是管状几何形状（例如，血管）中流动的一维降阶模型。在许多降阶模型中，模型方程是通过一些分析方法先验地描述的。在本文中，我们建议使用多尺度方法，它可以替代降阶模型，并且可以通过使用较少的基函数来表示降维建模（例如，使用一个基函数对应于一维近似）。

论文中考虑的数学模型由速度、压力和浓度方程组描述，其中流动由斯托克斯方程描述，输运由非均匀边界条件的非定常对流扩散方程描述在墙上（反应边界）。我们从非结构化网格上问题的有限元逼近开始，并将其用作二维和三维模型问题的参考解决方案。精细网格近似可在网格级别解析复杂的几何图形，并导致求解计算成本高昂的大型离散方程系统。为了减小离散系统的大小，我们开发了一种多尺度模型缩减技术，我们构建局部多尺度基函数以在粗网格上生成低维模型。建议的多尺度模型简化基于不连续伽辽金广义多尺度有限元方法 (DG-GMsGEM)。在针对流动问题的 DG-GMsFEM 中，我们首先为粗网格单元之间的每个界面构建快照空间以捕获可能的流动。为了减少快照空间的大小，我们通过局部谱问题的解决方案进行降维，并使用对应于最小特征值的特征向量作为粗网格上的近似的多尺度基函数。对于交通问题，我们为粗网格单元之间的每个界面构建多尺度基函数，并提供额外的基函数来捕获壁上的非均匀边界条件。最后，我们将针对二维和三维问题的三种测试几何进行一些数值模拟，以证明该方法的性能。