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首页> 外文期刊>Journal of Computational Physics >Reduced basis finite element heterogeneous multiscale method for high-order discretizations of elliptic homogenization problems
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Reduced basis finite element heterogeneous multiscale method for high-order discretizations of elliptic homogenization problems

机译：椭圆均质问题高阶离散化的约简有限元异质多尺度方法

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摘要

A new finite element method for the efficient discretization of elliptic homogenization problems is proposed. These problems, characterized by data varying over a wide range of scales cannot be easily solved by classical numerical methods that need mesh resolution down to the finest scales and multiscale methods capable of capturing the large scale components of the solution on macroscopic meshes are needed. Recently, the finite element heterogeneous multiscale method (FE-HMM) has been proposed for such problems, based on a macroscopic solver with effective data recovered from the solution of micro problems on sampling domains at quadrature points of a macroscopic mesh. Departing from the approach used in the FE-HMM, we show that interpolation techniques based on the reduced basis methodology (an offline-online strategy) allow one to design an efficient numerical method relying only on a small number of accurately computed micro solutions. This new method, called the reduced basis finite element heterogeneous multiscale method (RB-FE-HMM) is significantly more efficient than the FE-HMM for high order macroscopic discretizations and for three-dimensional problems, when the repeated computation of micro problems over the whole computational domain is expensive. A priori error estimates of the RB-FE-HMM are derived. Numerical computations for two and three dimensional problems illustrate the applicability and efficiency of the numerical method.
机译：提出了一种有效离散化椭圆均质化问题的新的有限元方法。这些问题的特点是数据在很大范围内变化，这些问题无法通过经典的数值方法轻松解决，而经典的数值方法需要将网格分辨率降低到最细的比例，因此需要能够在宏观网格上捕获解决方案的大规模分量的多尺度方法。近来，已经提出了一种针对此类问题的有限元异构多尺度方法（FE-HMM），该方法基于从宏观网格的正交点上的采样域上的微观问题的解中回收到的有效数据的宏观解算器。与FE-HMM中使用的方法不同，我们证明了基于简化基础方法（一种离线在线策略）的插值技术允许人们设计一种仅依靠少量精确计算的微解的有效数值方法。这种新方法称为简化基础有限元异质多尺度方法（RB-FE-HMM），在高阶宏观离散化和三维问题上，当重复计算微观问题时，其效率显着高于FE-HMM。整个计算域非常昂贵。得出RB-FE-HMM的先验误差估计。二维和三维问题的数值计算说明了数值方法的适用性和有效性。

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