Abstract: In this talk, we revisit the fundamentals of the Multiscale Hybrid-Mixed (MHM) method, initially proposed for a multiscale Laplace equation in [1] and further analyzed in [2,3]. The MHM method couples a global problem, defined on a skeleton mesh, with a collection of uncoupled local problems, each associated with the elements of a polytopal partition. These problems are derived from a static condensation process combined with a space decomposition applied to the hybrid formulation of the original problem. This decomposition ensures the well-posedness of the global and local problems by defining them as constrained problems. The second part of this talk introduces a novel variational methodology that bypasses the need for such a space decomposition, directly formulating the methods through elliptic local and global problems. This new approach forms the basis of the Multiscale Hybrid (MH) method, introduced in [4]. We present a comprehensive theoretical framework, numerical experiments, and a computational performance analysis, comparing both methods. [1] Harder, C., Paredes, D. and Valentin, F. "A Family of Multiscale Hybrid Mixed Finite Element Methods for the Darcy Equation with Rough Coefficients", Journal of Computational Physics, Vol. 245, pp. 107-130, 2013 [2] Araya, R., Harder, C., Paredes, D. and Valentin, F. "Multiscale Hybrid-Mixed Methods", SIAM Journal on Numerical Analysis, Vol. 51(6), pp. 3505-3531, 2013 [3] Barrenechea, G., Jaillet, F., Paredes, D., and Valentin, F. "The Multiscale Hybrid Mixed Method in General Polygonal Meshes", Numerische Mathematik, Vol. 145, pp. 197-237, 2020 [4] Barrenechea, G. R., Gomes, A. T. A., Paredes, D. "A Multiscale Hybrid Method". SIAM Journal on Scientific Computing, Vol. 46, pp. A1628-A1657, 2024