Bridging the Multiscale Hybrid-Mixed and Multiscale Hybrid High-Order methods

Théophile Chaumont-Frelet (1) , Alexandre Ern (2, 3) , Simon Lemaire (4) , Frédéric Valentin (5, 1)

1 ATLANTIS - Modélisation et méthodes numériques pour le calcul d'interactions onde-matière nanostructurée
2 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique
3 SERENA - Simulation for the Environment: Reliable and Efficient Numerical Algorithms
4 RAPSODI - Reliable numerical approximations of dissipative systems
5 LNCC / MCT - Laboratorio Nacional de Computação Cientifica [Rio de Janeiro]

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

Domaines: Analyse numérique [math.NA]

Théophile Chaumont-Frelet, Alexandre Ern, Simon Lemaire, Frédéric Valentin. Bridging the Multiscale Hybrid-Mixed and Multiscale Hybrid High-Order methods. 2021. ⟨hal-03235525v1⟩