Revisiting the robustness of the multiscale hybrid-mixed method: The face-based strategy

This work proposes a new finite element for the mixed multiscale hybrid method (MHM) applied to Poisson equation with highly oscillatory coefficients. Unlike original MHM method, bases are solution local Neumann problems driven by piecewise continuous polynomial interpolation on skeleton faces of macroscale mesh. As result, we prove optimal convergence refining face partition and leaving mesh macroelements fixed. property allows be resonance free under usual assumptions regularity. The numerical analysis also revisits complements approach proposed D. Paredes, F. Valentin H. Versieux (2017). Numerical experiments assess theoretical results.