非分离尺度与高对比度下有限维均匀化近似的局部化基
Localized bases for finite dimensional homogenization approximations with non-separated scales and high-contrast
AI总结：
针对具有L∞系数的散度型算子，提出一种不依赖遍历性或尺度分离的有限维近似方法，通过局部子域上求解椭圆PDE生成近似空间，并给出H1误差估计。
摘要：
我们构造了具有$L^\infty$系数的散度型算子解空间的有限维近似。我们的方法不依赖于遍历性或尺度分离的概念，而是依赖于这些算子的解空间在源项位于$L^2$的单位球而非$H^{-1}$的单位球时紧嵌入$H^1$的性质。近似空间通过在局部子域上求解椭圆PDE生成，其中源项对应于$H^2$的近似基。$H^1$误差估计表明，具有局部化到直径$\mathcal{O}(h^\alpha\ln \frac{1}{h})$（其中$\alpha\in [1/2,1)$）的子域上的基元素的$\mathcal{O}(h^{-d})$维空间，对于椭圆、抛物和双曲问题可达到$\mathcal{O}(h^{2-2\alpha})$的精度。对于高对比度介质，只要局部子域包含宽度为$\mathcal{O}(h^\alpha\ln \frac{1}{h})$的缓冲区域（其中介质对比度保持有界），该方法的精度得以保持。所提出的方法可以自然地推广到向量方程（如弹性动力学）。
英文摘要：
We construct finite-dimensional approximations of solution spaces of divergence form operators with $L^\infty$-coefficients. Our method does not rely on concepts of ergodicity or scale-separation, but on the property that the solution space of these operators is compactly embedded in $H^1$ if source terms are in the unit ball of $L^2$ instead of the unit ball of $H^{-1}$. Approximation spaces are generated by solving elliptic PDEs on localized sub-domains with source terms corresponding to approximation bases for $H^2$. The $H^1$-error estimates show that $\mathcal{O}(h^{-d})$-dimensional spaces with basis elements localized to sub-domains of diameter $\mathcal{O}(h^α\ln \frac{1}{h})$ (with $α\in [1/2,1)$) result in an $\mathcal{O}(h^{2-2α})$ accuracy for elliptic, parabolic and hyperbolic problems. For high-contrast media, the accuracy of the method is preserved provided that localized sub-domains contain buffer zones of width $\mathcal{O}(h^α\ln \frac{1}{h})$ where the contrast of the medium remains bounded. The proposed method can naturally be generalized to vectorial equations (such as elasto-dynamics).
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