Methods
PyMHM provides local/global constructions for MHM and related multiscale formulations. The tables identify the available formulations and their primary references. Consult the cited publications for mathematical definitions, approximation-space requirements and assumptions.
Use the Tutorials for the implementation workflow and the API for the supported operators, spaces and input contracts. The Gallery presents applications with their chosen data and discretizations.
MHM formulations
| Formulation | Local problem or model | Primary reference |
|---|---|---|
| Primal MHM | Scalar diffusion; primal local problems | Harder, Paredes and Valentin (2013) |
| Mixed H(div) MHM | Local flux–pressure mixed problems | Durán et al. (2019) |
| Petrov–Galerkin MHM | Residual-enriched scalar formulation | Fernando et al. (2023) |
| Scalar RAD MHM | Reaction–advection–diffusion | Harder et al. (2015); Araya et al. (2024) |
| Stokes–Brinkman MHM | Velocity–pressure local problems | Araya et al. (2017) |
| Oseen MHM | Flow with prescribed convection | Araya et al. (2021) |
| Primal elasticity MHM | Displacement local problems | Harder, Madureira and Valentin (2016) |
| GaLS elasticity MHM | Stabilized displacement–pressure local problems | Gomes, Pereira and Valentin (2024, preprint v1) |
| Mixed elasticity MHM | Stress, displacement and rotation local problems | Devloo et al. (2021) |
| Elastodynamic MHM | Time-dependent elasticity | Gomes et al. (2017) |
| Helmholtz MHM | Frequency-domain wave problems | Chaumont-Frelet and Valentin (2020) |
| Maxwell MHM | Time-domain electromagnetic problems | Lanteri et al. (2018) |
Related formulations
These formulations have their own local and global equations.
| Formulation | Primary reference |
|---|---|
| Robin Multiscale Hybrid method (MH) | Barrenechea, Gomes and Paredes (2024) |
| Three-field Multiscale Method (MH²M) | de Barros, Madureira and Valentin (2026, preprint v3) |
| Multiscale Hybrid High-Order method (MsHHO) | Cicuttin, Ern and Lemaire (2019); Chaumont-Frelet et al. (2022) |
Complementary strategies
| Strategy | Primary reference |
|---|---|
| MHM-USFEM: stabilized reaction–diffusion local problems | Martins, Pacazuca and Valentin (CILAMCE 2025) |
| Independent skeletal refinement | Paredes, Valentin and Versieux (2024) |
| Unfitted macro meshes with material-fitted skeletal partitions | Chaumont-Frelet, Paredes and Valentin (2026) |
| Flux reconstruction and associated estimators | Barrenechea et al. (2026) |
| Residual estimators and adaptive refinement | Araya et al. (2013); Araya, Rebolledo and Valentin (2021) |
The package also provides transient scalar integration and nested local/global providers. Their supported software interfaces are documented separately from the reference formulations above.
Citing
If you are using PyMHM or any of the supported methods, please cite the publications for the methods and strategies used, together with the PyMHM version. The Bibliography provides the complete bibliographic records, including versioned preprints.