论文标题
多尺度随机场的数值均质化修饰细胞问题的分析和数值研究
Analytical and numerical study of a modified cell problem for the numerical homogenization of multiscale random fields
论文作者
论文摘要
具有多尺度系数的部分微分方程的数值均质化的中心问题是准确计算有效数量的，例如均质系数。计算均质的系数需要解决局部纠正措施问题，然后再提高相关局部数据。计算均质系数的最幼稚方法是解决局部椭圆问题，该问题众所周知，该问题遭受了所谓的共振误差，该共振误差主导了多尺度计算中固有的所有其他误差。基于将指数校正项添加到标准的局部椭圆问题上的更有效的建模策略最近已被证明导致相对于局部几何形状的大小呈指数衰减的误差范围。与该方法的准确性和计算效率有关的问题先前已经在定期均质化的背景下解决了。本文涉及该修改后的椭圆校正问题的数学和数值研究扩展到随机均质化问题。特别是，我们假设一个固定的，千古的微结构，i）建立纠正方程的良好性，ii）分析来自模型中其他指数校正项的偏差（或系统误差）。提出了证实我们理论发现的数值结果。
A central question in numerical homogenization of partial differential equations with multiscale coefficients is the accurate computation of effective quantities, such as the homogenized coefficients. Computing homogenized coefficients requires solving local corrector problems followed by upscaling relevant local data. The most naive way of computing homogenized coefficients is by solving a local elliptic problem, which is known to suffer from the so-called resonance error dominating all other errors inherent in multiscale computations. A far more efficient modelling strategy, based on adding an exponential correction term to the standard local elliptic problem, has recently been proved to result in exponentially decaying error bounds with respect to the size of the local geometry. The questions in relation with the accuracy and computational efficiency of this approach has been previously addressed in the context of periodic homogenization. The present article concerns the extension of mathematical and numerical study of this modified elliptic corrector problem to stochastic homogenization problems. In particular, we assume a stationary, ergodic micro-structure and i) establish the well-posedness of the corrector equation, ii) analyse the bias (or the systematic error) originating from additional exponential correction term in the model. Numerical results corroborating our theoretical findings are presented.
