基于度量的升尺度方法
Metric based up-scaling
AI总结：
针对具有L∞系数的散度型椭圆算子，证明解在调和坐标下具有C^{1,α}可微性，并据此提出一种通过传递新度量和平均量的数值均匀化方法，适用于非遍历多尺度介质，同时可作为微分算子的压缩工具。
摘要：
我们考虑维度$n\geq 2$中具有$L^\infty$系数的散度型椭圆算子。尽管这些算子的解仅具有Hölder连续性，但我们证明它们在调和坐标下是可微的（$C^{1,α}$）。由此，数值均匀化可以扩展到介质在小尺度上不具有遍历性且以连续尺度谱为特征的情况，方法是将新度量以及传统的平均（均匀化）量从子网格尺度传递到计算尺度，并可以给出误差界。这种数值均匀化方法也可用作微分算子的压缩工具。
英文摘要：
We consider divergence form elliptic operators in dimension $n\geq 2$ with $L^\infty$ coefficients. Although solutions of these operators are only Hölder continuous, we show that they are differentiable ($C^{1,α}$) with respect to harmonic coordinates. It follows that numerical homogenization can be extended to situations where the medium has no ergodicity at small scales and is characterized by a continuum of scales by transferring a new metric in addition to traditional averaged (homogenized) quantities from subgrid scales into computational scales and error bounds can be given. This numerical homogenization method can also be used as a compression tool for differential operators.
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