Bridging the multiscale hybrid-mixed and multiscale hybrid high-order methods

We establish the equivalence between Multiscale Hybrid-Mixed (MHM) and Hybrid High-Order (MsHHO) methods for a variable diffusion problem with piecewise polynomial source term. Under idealized assumption that local problems defining multiscale basis functions are exactly solved, we prove holds general polytopal (coarse) meshes arbitrary approximation orders. also leverage interchange of properties to perform unified convergence analysis, as well improve on both methods.