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We study an abstract family of asymptotically degenerating variationalproblems. Those are natural generalisations of families of problems emergingupon application of a rescaled Floquet-Bloch-Gelfand transform to resolventproblems for high-contrast elliptic PDEs with highly oscillatory periodiccoefficients. An asymptotic analysis of these models leads us to a hierarchy ofapproximation results with uniform operator-type error estimates under variousassumptions, satisfied by specific examples. We provide approximations for theresolvents in terms of a certain `bivariate' operator which appears an abstractgeneralisation of the two-scale limit operators for highly oscillatoryhigh-contrast PDEs. The resulting approximating self-adjoint operator,providing tight operator error estimates, is the bivariate operator sandwichedby a connecting operator which for a broad class of periodic problemsspecialises to a new two-scale version of the classical Whittaker-Shannoninterpolation. An explicit description of the limit spectrum in the abstractsetting is provided, and new tight error estimates on the distance between theoriginal and limit spectra are established. Our generic approach allows us toreadily consider a wide class of asymptotically degenerating problems includingbut also going beyond high-contrast highly oscillatory PDEs. The obtainedresults are illustrated by various examples.

Uniform asymptotics for a family of degenerating variational problems and multiscale approximations with error estimates