Analysis of two-scale finite volume element method for elliptic problem

V. Ginting
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引用次数: 25

Abstract

In this paper we propose and analyze a class of finite volume element method for solving a second order elliptic boundary value problem whose solution is defined in more than one length scales. The method has the ability to incorporate the small scale behaviors of the solution on the large scale one. This is achieved through the construction of the basis functions on each element that satisfy the homogeneous elliptic differential equation. Furthermore, the method enjoys numerical conservation feature which is highly desirable in many applications. Existing analyses on its finite element counterpart reveal that there exists a resonance error between the mesh size and the small length scale. This result motivates an oversampling technique to overcome this drawback. We develop an analysis of the proposed method under the assumption that the coefficients are of two scales and periodic in the small scale. The theoretical results are confirmed experimentally by several convergence tests. Moreover, we present an application of the method to flows in porous media.
椭圆问题的二尺度有限体积元法分析
本文提出并分析了一类求解具有多个长度尺度的二阶椭圆型边值问题的有限体积元方法。该方法具有将解的小尺度行为结合到大尺度行为上的能力。这是通过在满足齐次椭圆微分方程的每个元素上构造基函数来实现的。此外,该方法还具有数值守恒的特点,这在许多应用中是非常理想的。现有的有限元分析表明,网格尺寸与小长度尺度之间存在共振误差。这个结果激发了过采样技术来克服这个缺点。在小尺度下,假设系数具有两个尺度和周期性,对所提出的方法进行了分析。通过若干收敛性实验验证了理论结果。此外,我们还介绍了该方法在多孔介质流动中的应用。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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