Adaptive finite element heterogeneous multiscale method for homogenization problems
Adaptive finite element heterogeneous multiscale method for homogenization problems
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DOI：
10.1016/j.cma.2010.06.012
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发表时间：
2011-09
影响因子：
7.2
通讯作者：
A. Abdulle;A. Nonnenmacher
中科院分区：
文献类型：
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作者：
A. Abdulle;A. Nonnenmacher
在本文中，我们提出了通过有限元异构多尺度方法离散的椭圆均匀化问题的后验误差分析。与标准有限元方法不同，我们的离散方案依赖于宏观和微观有限元。所需的宏观解决方案是通过基于微观数据的适当平均程序获得的。由于事先无法获得宏观数据（例如宏观扩散张量），因此必须定义适当的误差指标来设计自适应方法。我们表明，可以定义仅基于可用的宏观和微观解决方案（用于计算实际宏观解决方案）的指标，从而实现可靠且高效的宏观网格细化策略。上限和下限的相应后验估计是在能量范数中导出的。在均匀振荡张量的情况下，我们恢复了应用于均质问题的有限元方法的基于标准残差的后验误差估计。数值实验验证了自适应多尺度方法的效率和可靠性。
In this paper we present an a posteriori error analysis for elliptic homogenization problems discretized by the finite element heterogeneous multiscale method. Unlike standard finite element methods, our discretization scheme relies on macro- and microfinite elements. The desired macroscopic solution is obtained by a suitable averaging procedure based on microscopic data. As the macroscopic data (such as the macroscopic diffusion tensor) are not available beforehand, appropriate error indicators have to be defined for designing adaptive methods. We show that such indicators based only on the available macro- and microsolutions (used to compute the actual macrosolution) can be defined, allowing for a macroscopic mesh refinement strategy which is both reliable and efficient. The corresponding a posteriori estimates for the upper and lower bound are derived in the energy norm. In the case of a uniformly oscillating tensor, we recover the standard residual-based a posteriori error estimate for the finite element method applied to the homogenized problem. Numerical experiments confirm the efficiency and reliability of the adaptive multiscale method.