An H(div;Ω)-conforming flux reconstruction for the multiscale hybrid-mixed method

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Barrenechea, Gabriel R. and Martins, Larissa and Pereira, Weslley and Valentin, Frédéric (2026) An H(div;Ω)-conforming flux reconstruction for the multiscale hybrid-mixed method. Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal, 24 (2). pp. 399-428. ISSN 1540-3459 (https://doi.org/10.1137/24M1673073)

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Abstract

The Multiscale Hybrid-Mixed (MHM) method is a multiscale finite element method based on a hybrid weak formulation, originally proposed for problems linked to flow in porous media. Its starting point is a variational formulation that guarantees that the flux variable is H(div;Ω)-conforming. This property is lost when the local problems, in their elliptic form, are discretized using primal finite element methods. In this work, we close that gap by proposing and analyzing a new flux reconstruction for the MHM method, computed element-wise on submeshes, that belongs to H(div;Ω). The reconstruction converges optimally in the L2(Ω)−norm. Furthermore, the divergence of the reconstructed flow is the piecewise continuous polynomial projection of the source term onto submeshes and thus it also converges optimally in the L2(Ω)−norm. As a by-product of the reconstruction technique, a fully computable a posteriori error estimator is presented and analyzed. These theoretical results are validated experimentally via numerical computations.

ORCID iDs

Barrenechea, Gabriel R. ORCID logoORCID: https://orcid.org/0000-0003-4490-678X, Martins, Larissa, Pereira, Weslley and Valentin, Frédéric;
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