Contrast independent localization of multiscale problems
For researchers in multiscale simulation, this work provides a theoretical and practical solution to a known bottleneck (contrast-dependent error) in localized multiscale methods.
The paper addresses the contrast-dependent localization error in multiscale methods and presents a novel interpolation operator for two-valued coefficients that achieves contrast-independent localization error under physically justified assumptions, validated by numerical experiments.
The accuracy of many multiscale methods based on localized computations suffers from high contrast coefficients since the localization error generally depends on the contrast. We study a class of methods based on the variational multiscale method, where the range and kernel of a quasi-interpolation operator defines the method. We present a novel interpolation operator for two-valued coefficients and prove that it yields contrast independent localization error under physically justified assumptions on the geometry of inclusions and channel structures in the coefficient. The idea developed in the paper can be transferred to more general operators and our numerical experiments show that the contrast independent localization property follows.