Robust Numerical Upscaling of Elliptic Multiscale Problems at High Contrast

TL;DR

This paper introduces a new contrast-independent numerical upscaling method for elliptic multiscale problems with high contrast, improving convergence and robustness over existing schemes.

Contribution

It develops a novel localizable orthogonal decomposition framework using stable quasi-interpolation operators that achieve contrast-independent optimal convergence.

Findings

Achieves contrast-independent optimal convergence.

Reduces pre-asymptotic effects in high-contrast problems.

Demonstrates flexibility for various quasi-interpolation operators.

Abstract

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.