Foundations of the MHM method

Abstract

An abstract setting for the construction and analysis of the Multiscale Hybrid-Mixed (MHM for short) method is proposed. We review some of the most recent developments from this standpoint, and establish relationships with the classical lowest-order Raviart-Thomas element and the primal hybrid method, as well as with some recent multiscale methods. We demonstrate the reach of the approach by revisiting the wellposedness and error analysis of the MHM method applied to the Laplace problem. In the process, we devise new theoretical results for this model.