具有反应边界薄区域流动输运问题的多尺度降维
Multiscale dimension reduction for flow and transport problems in thin domain with reactive boundaries
摘要：
在本文中，我们考虑薄域中的流动和运输问题。本文中考虑的数学模型由速度、压力和浓度的方程组描述，其中流动由斯托克斯方程描述，输送由非定常对流-扩散方程描述，在壁上具有非齐次边界条件（反应边界）。我们从非结构网格上的问题的有限元近似开始，并将其用作二维和三维模型问题的参考解决方案。精细网格近似在网格水平上解决复杂的几何形状，并导致求解计算昂贵的大型离散方程组。为了减少的离散系统的大小，我们开发了一个多尺度模型简化技术，在那里我们构建本地多尺度基函数生成一个低维模型上的粗网格。所提出的多尺度模型降阶是基于不连续Galerkin广义多尺度有限元法（DG-GMsGEM）。在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。对于输运问题，我们构造多尺度基函数的粗网格单元之间的每个接口，并提出额外的基函数捕捉非齐次边界条件的墙壁。最后，我们将提出一些数值模拟的三个测试几何的二维和三维问题，以证明该方法的性能。
英文摘要：
In this paper, we consider flow and transport problems in thin domains. 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
