On the robustness of multiscale hybrid-mixed methods

Abstract:

In this work we prove uniform convergence of the Multiscale Hybrid-Mixed (MHM for short) finite element method for second-order elliptic problems with rough periodic coefficients. The MHM method is shown to avoid resonance errors without adopting oversampling techniques. In particular, we establish that the discretization error for the primal variable in the broken $H^1$ and $L^2$ norms are $O(h + \varepsilon ^\delta )$ and $O(h^2 + h \varepsilon ^\delta )$, respectively, and for the dual variable it is $O(h + \varepsilon ^\delta )$ in the $H(\operatorname {div};\cdot )$ norm, where $0<\delta \leq 1/2$ (depending on regularity). Such results rely on sharpened asymptotic expansion error estimates for the elliptic models with prescribed Dirichlet, Neumann or mixed boundary conditions.