A Sub-Grid Structure Enhanced Discontinuous Galerkin Method for Multiscale Diffusion and Convection-Diffusion Problems
A Sub-Grid Structure Enhanced Discontinuous Galerkin Method for Multiscale Diffusion and Convection-Diffusion Problems
复制标题
解决多尺度扩散和对流扩散问题的子网格结构增强间断伽辽金法
DOI：
10.4208/cicp.071211.070912a
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发表时间：
2013
影响因子：
3.7
通讯作者：
W. Leung
中科院分区：
文献类型：
--
作者：
Eric T. Chung;W. Leung
在本文中，我们提出了一种有效的计算方法来求解高度非均质介质中的扩散和对流扩散问题以及对流主导的扩散问题。众所周知，对这些问题进行数值计算需要大量的计算机内存和时间。然而，这些问题的解决方案通常包含一个粗糙的组成部分，它通常是感兴趣的数量，可以用少量的自由度来表示。有许多方法的目的是计算粗分量，而不解决解决方案的全部细节。本文提出的方法属于内罚不连续伽辽金法的框架，是一类求解偏微分方程数值解的有效而精确的方法。该方法的一个显著特点是解空间包含两个分量，即传统方法中对粗分量给出多尺度逼近的粗空间和包含解的子网格结构的多尺度空间，这对粗分量的计算至关重要。此外，还证明了该方法的稳定性。数值结果表明，该方法可以准确地捕捉到高度非均质介质中问题解的粗糙行为，以及对流主导问题的边界层和内层。AMS学科分类：65M12、65M60
In this paper, we present an efficient computational methodology for dif- fusion and convection-diffusion problems in highly heterogeneous media as well as convection-dominated diffusion problem. It is well known that the numerical compu- tation for these problems requires a significant amount of computer memory and time. Nevertheless, the solutions to these problems typically contain a coarse component, which is usually the quantity of interest and can be represented with a small num- ber of degrees of freedom. There are many methods that aim at the computation of the coarse component without resolving the full details of the solution. Our proposed method falls into the framework of interior penalty discontinuous Galerkin method, which is proved to be an effective and accurate class of methods for numerical solu- tions of partial differential equations. A distinctive feature of our method is that the solution space contains two components, namely a coarse space that gives a polyno- mial approximation to the coarse component in the traditional way and a multiscale space which contains sub-grid structures of the solution and is essential to the com- putation of the coarse component. In addition, stability of the method is proved. The numerical results indicate that the method can accurately capture the coarse behavior of the solution for problems in highly heterogeneous media as well as boundary and internal layers for convection-dominated problems. AMS subject classifications: 65M12, 65M60