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

TL;DR

This paper proves the equivalence of two multiscale numerical methods for variable diffusion problems, enabling unified analysis and improvements for both approaches on complex meshes.

Contribution

It establishes the theoretical equivalence between MHM and MsHHO methods, allowing for unified convergence analysis and method enhancements.

Findings

Proves equivalence between MHM and MsHHO methods.

Enables unified convergence analysis.

Improves both methods based on the equivalence.

Abstract

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 term. 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. We also leverage the interchange of properties to perform a unified convergence analysis, as well as to improve on both methods.