Multi-scale Discontinuous Galerkin Method for Solving Elliptic Problems with Curvilinear Unidirectional Rough Coefficients
Multi-scale Discontinuous Galerkin Method for Solving Elliptic Problems with Curvilinear Unidirectional Rough Coefficients
复制标题
求解具有曲线单向粗糙系数的椭圆问题的多尺度间断伽辽金法
作者：
Yifan Zhang;Wei Wang;J. Guzmán;Chi
本文通过选择一个特殊的非多项式逼近空间，对具有曲线单向粗糙系数的二阶椭圆型问题提出了一种多尺度间断Galerkin（DG）方法.该方法的关键在于将微分算子的局部振荡特征引入到近似空间中，从而在不需要求解最细尺度的情况下捕获多尺度解。粗糙系数的单向性使我们能够显式地构造DG非多项式逼近空间的基函数，从而大大提高了算法的效率。详细的二维二阶DG方法的误差估计，并讨论了如何构造这样的非多项式基的一般指导。数值算例验证了算法的有效性.
In this paper, we propose a multi-scale discontinuous Galerkin (DG) method for second-order elliptic problems with curvilinear unidirectional rough coefficients by choosing a special non-polynomial approximation space. The key ingredient of the method lies in the incorporation of the local oscillatory features of the differential operators into the approximation space so as to capture the multi-scale solutions without having to resolve the finest scales. The unidirectional feature of the rough coefficients allows us to construct the basis functions of the DG non-polynomial approximation space explicitly, thereby greatly increasing the algorithm efficiency. Detailed error estimates for two-dimensional second-order DG methods are derived, and a general guidance on how to construct such non-polynomial basis is discussed. Numerical examples are also presented to validate and demonstrate the effectiveness of the algorithm.