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We present a new approach to the numerical upscaling for elliptic problemswith rough diffusion coefficient at high contrast. It is based on thelocalizable orthogonal decomposition of $H^1$ into the image and the kernel ofsome novel stable quasi-interpolation operators with local $L^2$-approximationproperties, independent of the contrast. We identify a set of sufficientassumptions on these quasi-interpolation operators that guarantee in principleoptimal convergence without pre-asymptotic effects for high-contrastcoefficients. We then give an example of a suitable operator and establish theassumptions for a particular class of high-contrast coefficients. So far thisis not possible without any pre-asymptotic effects, but the optimal convergenceis independent of the contrast and the asymptotic range is largely improvedover other discretisation schemes. The new framework is sufficiently flexibleto allow also for other choices of quasi-interpolation operators and thepotential for fully robust numerical upscaling at high contrast.

Robust Numerical Upscaling of Elliptic Multiscale Problems at High Contrast