Robust Numerical Upscaling of Elliptic Multiscale Problems at High Contrast
This work addresses the challenge of robust numerical upscaling for high-contrast multiscale problems, which is important for applications in porous media and composite materials.
The paper introduces a new numerical upscaling method for elliptic multiscale problems with high-contrast diffusion coefficients, achieving optimal convergence independent of contrast and significantly improving the asymptotic range over existing schemes.
We present a new approach to the numerical upscaling for elliptic problems with rough diffusion coefficient at high contrast. It is based on the localizable orthogonal decomposition of $H^1$ into the image and the kernel of some novel stable quasi-interpolation operators with local $L^2$-approximation properties, independent of the contrast. We identify a set of sufficient assumptions on these quasi-interpolation operators that guarantee in principle optimal convergence without pre-asymptotic effects for high-contrast coefficients. We then give an example of a suitable operator and establish the assumptions for a particular class of high-contrast coefficients. So far this is not possible without any pre-asymptotic effects, but the optimal convergence is independent of the contrast and the asymptotic range is largely improved over other discretisation schemes. The new framework is sufficiently flexible to allow also for other choices of quasi-interpolation operators and the potential for fully robust numerical upscaling at high contrast.