A Petrov-Galerkin multiscale hybrid-mixed method for the Darcy equation on polytopes

In this work, we propose a new method called Petrov-Galerkin multiscale hybrid-mixed (PGMHM for short) as a variant of the multiscale hybrid-mixed (MHM) method. Its construction starts from a Petrov-Galerkin formulation for the Lagrange multiplier space, defined by enriching the trial spaces with residual-based functions on the partition faces. As a result, jump terms are added to the original MHM method, which penalizes the lack of conformity of MHM numerical solutions. As a consequence of space enrichment, the method induces local post-processing of the numerical solution that incorporates the model's physical aspects and preserves the exact solution's local conservation properties. Numerical experiments validate the theoretical results and verify the accuracy of PGMHM on highly heterogeneous problems.