论文标题
一个椭圆形的局部问题，具有数值均质的共振误差的指数衰减
An elliptic local problem with exponential decay of the resonance error for numerical homogenization
论文作者
论文摘要
数值多尺度方法通常依赖于宏观模型和微观模型之间的某些耦合。该模型中缺少宏观模型，因为有效量（例如均质的材料系数或通量）不完整。这些有效数据需要通过运行本地微观模拟，然后进行微观信息的局部平均来计算。受经典均质化理论的促进，通常使用局部椭圆细胞问题来计算宏模型中缺失的均质系数。这样的考虑导致一阶错误$ o（\ varepsilon/δ）$，其中$ \ varepsilon $代表微观变化的波长和$δ$是微观模拟框的大小。此错误称为“共振误差”，源自微问题中使用的边界条件，通常以多尺度数值方法中的所有其他错误主导。共振误差的最佳衰减仍然是一个空旷的问题，尽管在过去的二十年中，已经提出了几种有趣的方法来降低边界效果。在本文中，作为解决此问题的尝试，我们提出了一种计算高效，完全椭圆形的方法，并具有共振误差的指数衰减。
Numerical multiscale methods usually rely on some coupling between a macroscopic and a microscopic model. The macroscopic model is incomplete as effective quantities, such as the homogenized material coefficients or fluxes, are missing in the model. These effective data need to be computed by running local microscale simulations followed by a local averaging of the microscopic information. Motivated by the classical homogenization theory, it is a common practice to use local elliptic cell problems for computing the missing homogenized coefficients in the macro model. Such a consideration results in a first order error $O(\varepsilon/δ)$, where $\varepsilon$ represents the wavelength of the microscale variations and $δ$ is the size of the microscopic simulation boxes. This error, called "resonance error", originates from the boundary conditions used in the micro-problem and typically dominates all other errors in a multiscale numerical method. Optimal decay of the resonance error remains an open problem, although several interesting approaches reducing the effect of the boundary have been proposed over the last two decades. In this paper, as an attempt to resolve this problem, we propose a computationally efficient, fully elliptic approach with exponential decay of the resonance error.
