A Petrov-Galerkin multiscale hybrid-mixed method for the Darcy equantion on polytopes - Martins, Larissa de Souza (2019)
In this work, we modify the Multiscale Hybrid-Mixed (MHM) method and propose a new multiscale finite element method, called Petrov-Galerkin Multiscale Hybrid-Mixed, PGMHM for short. Its construction starts from a Petrov-Galerkin formulation for the Lagrange multiplier space, defined by enriching the Lagrange multiplier polynomial trial space with residual-based functions restricted to 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 they are residual-based, the additional terms preserve the consistency of the original MHM method. Also, as a consequence of space enrichment, the method induces a local postprocessing of the numerical solution. Such an enriched solution belongs to a “richer” space induced by a discrete Lagrange multiplier space that incorporates physical aspects of the model and preserves the local conservation properties of the exact solution. We prove that the PGMHM method achieves superconvergence properties assuming local regularity, as the original MHM method. Numerical experiments validate the theoretical results and verify the accuracy of PGMHM on highly heterogeneous problems.