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

This work proposes a new finite element for the multiscale hybrid-mixed method (MHM) applied to the Poisson equation with highly oscillatory coefficients. Unlike the original MHM method, multiscale bases are the solution to local Neumann problems driven by piecewise continuous polynomial interpolation on the skeleton faces of the macroscale mesh. As a result, we prove the optimal convergence of MHM by refining the face partition and leaving the mesh of macroelements fixed. This property allows the MHM method to be resonance free under the usual assumptions of local regularity. The numerical analysis of the method also revisits and complements the original approach proposed by D. Paredes, F. Valentin and H. Versieux (2017). Numerical experiments assess the new theoretical results.& COPY; 2023 Elsevier B.V. All rights reserved.