Highlights
The poster “A Stabilized Multiscale Hybrid-Mixed Method for Reaction-Dominated Models” was
presented by PhD student Juan Pacazuca at the Inria-Brasil (hybrid) Workshop on HPC, held at LNCC,
Petrópolis, Brazil, on April 15–16, 2026. The event brought together French and Brazilian researchers
to foster collaboration in high-performance computing, artificial intelligence, scientific computing,
and data science. The work highlights ongoing research on advanced numerical methods for challenging
multiscale problems.
During the workshop “Taming the PDEs: Tailored Methods, Multiscale Approaches,
and Real-World Application”, held on March 9–13, 2026 as part of the Junior Trimester
Program at the Hausdorff Research Institute for Mathematics in Bonn, Larissa Martins
presented the talk “H(div, Ω)-Conforming Multiscale Hybrid-Mixed Methods for
Elasticity with Weak and Strong Symmetry.” The presentation introduced new
elementwise reconstruction strategies within the Multiscale Hybrid-Mixed (MHM)
framework for linear elasticity that produce H(div; Ω)-conforming stress tensors
while enforcing either weak or strong symmetry. The proposed approach restores local
conservation and improves the quality of stress approximations obtained from continuous
Galerkin local solvers. Theoretical analysis proves optimal convergence of the
reconstructed stress in the H(div; Ω)-norm, while numerical experiments demonstrate
the robustness and effectiveness of the method in multilayer elasticity problems
relevant to real-world applications such as faulted subsurface reservoirs.
🎥 Presentation recording: https://youtu.be/noJqq1eotX4
At the XLVI Ibero-Latin American Congress on Computational Methods in Engineering (CILAMCE 2025),
Juan Felipe Pacazuca Santiago presented the work “A Multiscale Hybrid-Mixed Method with Local Stabilization.”
The talk introduced the MHM-UNUSUAL method, a new approach that combines the Multiscale Hybrid-Mixed (MHM)
framework with the Unusual Stabilized Finite Element Method (UNUSUAL). The proposed strategy improves
the approximation of multiscale basis functions in challenging scenarios, such as reaction-dominated and
highly heterogeneous problems. By incorporating stabilization terms in the local problems, the method
mitigates spurious oscillations and allows the use of coarser local meshes, reducing computational cost
while maintaining accuracy. Numerical experiments on boundary layer problems and the SPE-10 benchmark demonstrate
the potential of the approach for efficient and reliable multiscale simulations.
Larissa Martins and Diego Paredes presented recent advances on Multiscale
Hybrid-Mixed (MHM) methods at the workshop Reduced-Order Modeling for Complex
Engineering Problems: From Analysis to Practical Implementation, held from January 29
to February 7, 2025, at the Institute for Mathematical and Statistical Innovation (IMSI), in Chicago, USA.
The event focused on numerical simulation of engineering problems in complex and
heterogeneous media, highlighting multiscale methods, reduced-order modeling,
and practical implementation in industrial contexts.
Diego Paredes (Universidad de Concepción) delivered the talk "Multiscale
Hybrid Methods: Theoretical Foundations and Computational Analysis". In a
lightning talk, Larissa Martins presented "An H(div, Ω)-conforming flux
reconstruction for the MHM method".
This presentation was delivered at the mini-symposium New Challenges in the Numerical
Simulation of Partial Differential Equations (MS08) during CNMAC 2024, held on September
19–20 in Porto de Galinhas by Frédéric Valentin. The talk presented recent advances in the Multiscale Hybrid-Mixed (MHM)
method, focusing on the recovery of optimal convergence for the dual variable in H(div, Ω) through
a local post-processing strategy applied to the two-level MHM formulation. The approach restores
key properties of the original one-level method while maintaining computational efficiency. Additionally,
the work introduces a fully computable a posteriori error estimator based on equilibrated flux techniques,
providing a reliable tool for assessing the accuracy of multiscale simulations on coarse meshes.
This five-lecture course took place as part of the training program of L'École Doctorale en Sciences Fondamentales et Appliquées, at Université Côte d’Azur. The course presented an overview of the origin of the MHM method as well as the MHM's basic theory and practical aspects of the implementation of the MHM method, including practical numerical simulations using the MultiScale FEM Library (MSL).
We proposed a new finite element for the mixed multiscale hybrid method (MHM) applied to the Poisson equation with highly oscillatory coefficients. Unlike the original MHM method, multiscale bases are the solution to local Neumann problems driven by piecewise continuous polynomial interpolation on the skeleton faces of the macroscale mesh. As a result, we prove the optimal convergence of MHM by refining the face partition and leaving the mesh of macroelements fixed. This property allows the MHM method to be resonance free under the usual assumptions of local regularity.
Larissa Martins was the first place winner of the SBMAC Odelar Leite Linhares Award, Masters category (2021), for her dissertation "A Petrov-Galerkin Multiscale Hybrid-Mixed Method for the Darcy Equation on Polytopes". Larissa was supervised by Professors Frédéric Valentin (LNCC) and Honorio Fernando (UFF).
The research project EOLIS — Efficient Off-Line Numerical Strategies for Multi-Query Problems —, formed by researchers and students from the IPES and NUMA Research Groups at LNCC, and coordinated in Brazil by Frédéric Valentin, was selected in the new Math-Amsud 2020 call. STIC-AmSud and Math-AmSud are regional cooperation programs between France, Argentina, Bolivia, Brazil, Chile, Colombia, Ecuador, Paraguay, Peru, Uruguay and Venezuela. The objective is to implement joint projects and strengthen collaboration in the areas of Science and Information and Communication Technology, in the first, and in Mathematics, in the second. The approved projects last two years and involve at least two South American countries and one or more teams of French scientists.
The EOLIS project will receive funding from CAPES during the period 2021-2022, including post-graduate and post-doctoral scholarships. International collaboration involves INRIA from France, Universidad de Concepción and Universidad Católica de la Santísima Concepción from Chile, in addition to LNCC.
"The EOLIS project aims to develop new high-performance numerical and computational algorithms for the resolution of physical / biological models, which require a huge range of computational simulations due to large variations in the model data. A typical application is the numerical approximation of model solutions based on stochastic differential equations. This project strengthens the cooperation between leading research groups in scientific computing in France, Chile and Brazil and constitutes a natural development of the PHOTOM project ( Math-Amsud 2018-2020). ", emphasizes Frédéric Valentin.
Frederic Valentin has been awarded a grant in FAPERJ's
"Scientist of Our State 2020 - CNE" Program. His project aims to develop and
mathematically analyze new multi-scale finite element methods,
which give rise to new computational algorithms adapted to new generations
of massively parallel computers. Such algorithms aim to solve systems
of differential equations with heterogeneous and/or singularly disturbed
coefficients.
We proposed a new family of polytopal finite elements for the MHM method applied to the linear elasticity model in bidimensional and tridimensional problems. We extended the construction and the analysis of the MHM method proposed in other pieces of work that approached these problems using only simplx finite elements. We proved optimal convergence in the L2 norm using these polytopal finite elements. We also addressed computational aspects of this new family of methods, developed for running in parallel with different configurations, and we illustrated by means of computational experiments how to balance the execution time and the memory consumption in search for a certain approximation order. More information.
The figures above show the x and z components of displacement obtained for a linear elasticity model with the MHM method in a domain with null displacement in the bottom boundary and null traction in the remainder boundaries. The domain is divided into 16 layers according to the HPC4e benchmark, with layers 4 and 12 containing saturated clay and the other layers filled with the original materials from the benchmark. The source is the piecewise weight, using gravity acceleration of 9.8 m/s2. The employed mesh is rectangular throughout the domain (note how the layers "cross" the elements of the mesh), except for the top boundary, in which elements with an average of 35 edges are used for capturing the topography.