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We establish the equivalence between the Multiscale Hybrid-Mixed (MHM) andthe Multiscale Hybrid High-Order (MsHHO) methods for a variable diffusionproblem with piecewise polynomial source term. Under the idealized assumptionthat the local problems defining the multiscale basis functions are exactlysolved, we prove that the equivalence holds for general polytopal (coarse)meshes and arbitrary approximation orders. We also leverage the interchange ofproperties to perform a unified convergence analysis, as well as to improve onboth methods.

Bridging the Multiscale Hybrid-Mixed and Multiscale Hybrid High-Order methods