Capturing Small Scales in Elliptic Problems Using a Residual-Free Bubbles Finite Element Method
Capturing Small Scales in Elliptic Problems Using a Residual-Free Bubbles Finite Element Method
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DOI：
10.1137/s1540345902411402
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发表时间：
2003
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影响因子：
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通讯作者：
G. Sangalli
中科院分区：
文献类型：
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作者：
G. Sangalli
本文研究了求解二阶快速变系数椭圆方程的无残余气泡（RFB）有限元法。RFB技术与Hou、Wu、Cai等人提出的多尺度有限元法（MsFEM）密切相关。Comp., 68 (1999), pp. 913-943)以及在求解这类偏微分方程的工程文献中非常常见的升级程序。我们还介绍了基于宏气泡的RFB方法的一种变体，称为无残差宏气泡（RFMB）方法，该方法可以给出更精确的数值解。在周期系数的情况下，我们能够证明方法的先验误差估计。最后，我们对模型问题的数值方法进行了验证。
In this work we study the residual-free bubbles (RFB) finite element method for solv- ing second order elliptic equations with rapidly varying coefficients. The RFB technique is closely related to both the multiscale finite element method (MsFEM) introduced by Hou, Wu, and Cai (Math. Comp., 68 (1999), pp. 913-943) and the upscaling procedures which are very common in the engineering literature for solving this kind of partial differential equation. We also introduce a vari- ation of the RFB method, based on macrobubbles and referred to as the residual-free macrobubbles (RFMB) method, which gives more accurate numerical solutions. In the case of periodic coefficients we are able to prove a priori error estimates for the methods. Eventually, we test the numerical methods on model problems.