论文标题
基于LOD技术的多尺度元素的组合有限元方法，用于多尺度椭圆形问题
A combined multiscale finite element method based on the LOD technique for the multiscale elliptic problems with singularities
论文作者
论文摘要
在本文中，我们使用局部正交分解（LOD）技术构建了组合的多尺度有限元法（MSFEM），以解决在计算域的某些特殊部分中可能具有奇异性的多尺度问题。例如，在模拟通过提取井驱动的高度异质的多孔介质传输的稳定流动中，奇异性位于近孔区域。合并方法的基本思想是将传统的有限元方法（FEM）直接在域的有问题部分的细分网格上使用，并在另一部分的粗网格上使用基于LOD的MSFEM。关键点是如何定义元素在粗糙和细网界附近的元素的基础函数，这需要细致的处理。所提出的方法具有传统的FEM和基于LOD的MSFEM的优势，该方法的使用量比标准FEM少得多，并且可能比基于LOD的MSFEM更准确。对于高度变化的系数进行了误差分析，没有任何规模分离或周期性的假设。 {带有周期性和随机振荡系数的数值示例}，以及L形域上的多尺度问题，以及具有高对比度通道或良好性的多尺度问题，以证明该方法的效率和准确性。
In this paper, we construct a combined multiscale finite element method (MsFEM) using the Local Orthogonal Decomposition (LOD) technique to solve the multiscale problems which may have singularities in some special portions of the computational domain. For example, in the simulation of steady flow transporting through highly heterogeneous porous media driven by extraction wells, the singularities lie in the near-well regions. The basic idea of the combined method is to utilize the traditional finite element method (FEM) directly on a fine mesh of the problematic part of the domain and using the LOD-based MsFEM on a coarse mesh of the other part. The key point is how to define local correctors for the basis functions of the elements near the coarse and fine mesh interface, which require meticulous treatment. The proposed method takes advantages of the traditional FEM and the LOD-based MsFEM, which uses much less DOFs than the standard FEM and may be more accurate than the LOD-based MsFEM for problems with singularities. The error analysis is carried out for highly varying coefficients, without any assumptions on scale separation or periodicity. {Numerical examples with periodic and random highly oscillating coefficients}, as well as the multiscale problems on the L-shaped domain, and multiscale problems with high-contrast channels or well-singularities are presented to demonstrate the efficiency and accuracy of the proposed method.
