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Bibliography

Cite the publications for the methods used in a calculation, together with the PyMHM version. Methods lists the available constructions and their references. Journal articles and versioned preprints are identified separately.

Original MHM formulations and analyses

Harder, Paredes and Valentin (2013): The original Darcy family

Christopher Harder, Diego Paredes, and Frédéric Valentin (2013). A family of Multiscale Hybrid-Mixed finite element methods for the Darcy equation with rough coefficients, Journal of Computational Physics 245, 107–130. DOI: 10.1016/j.jcp.2013.03.019.

Araya et al. (2013): A priori and a posteriori analysis

Rodolfo Araya, Christopher Harder, Diego Paredes, and Frédéric Valentin (2013). Multiscale Hybrid-Mixed Method, SIAM Journal on Numerical Analysis 51(6), 3505–3531. DOI: 10.1137/120888223.

Harder and Valentin (2016): Abstract foundations

Christopher Harder and Frédéric Valentin (2016). Foundations of the MHM Method, in Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations, Lecture Notes in Computational Science and Engineering 114, Springer. DOI: 10.1007/978-3-319-41640-3_13.

Paredes, Valentin and Versieux (2017): Periodic-coefficient robustness

Diego Paredes, Frédéric Valentin, and Henrique M. Versieux (2017). On the robustness of multiscale hybrid-mixed methods, Mathematics of Computation 86(304), 525–548. DOI: 10.1090/mcom/3108.

Durán et al. (2019): Mixed local Darcy solvers

Omar Durán, Philippe R. B. Devloo, Sônia M. Gomes, and Frédéric Valentin (2019). A multiscale hybrid method for Darcy’s problems using mixed finite element local solvers, Computer Methods in Applied Mechanics and Engineering 354, 213–244. DOI: 10.1016/j.cma.2019.05.013.

Cicuttin, Ern and Lemaire (2019): Original MsHHO construction

Matteo Cicuttin, Alexandre Ern and Simon Lemaire (2019). A Hybrid High-Order Method for Highly Oscillatory Elliptic Problems, Computational Methods in Applied Mathematics 19(4), 723–748. DOI: 10.1515/cmam-2018-0013.

Chaumont-Frelet et al. (2022): The MHM–MsHHO connection

Théophile Chaumont-Frelet, Alexandre Ern, Simon Lemaire, and Frédéric Valentin (2022). Bridging the Multiscale Hybrid-Mixed and Multiscale Hybrid High-Order Methods, ESAIM: M2AN 56, 261–285. DOI: 10.1051/m2an/2021082.

Paredes, Valentin and Versieux (2024): Face-based robustness

Diego Paredes, Frédéric Valentin, and Henrique M. Versieux (2024). Revisiting the robustness of the multiscale hybrid-mixed method: The face-based strategy, Journal of Computational and Applied Mathematics 436, 115415. DOI: 10.1016/j.cam.2023.115415.

Chaumont-Frelet, Paredes and Valentin (2022, preprint v1): Unfitted flux approximation

Théophile Chaumont-Frelet, Diego Paredes, and Frédéric Valentin (2022). Flux approximation on unfitted meshes and application to multiscale hybrid-mixed methods, preprint, HAL version 1, 31 October 2022. HAL: hal-03834748v1.

Barrenechea et al. (2026): H(div) flux reconstruction

Gabriel R. Barrenechea, Larissa Martins, Weslley Pereira, and Frédéric Valentin (2026). An H(div; Ω)-Conforming Flux Reconstruction for the Multiscale Hybrid-Mixed Method, Multiscale Modeling & Simulation 24(2), 399–428. DOI: 10.1137/24M1673073.

Chaumont-Frelet, Paredes and Valentin (2026): Unfitted flux approximation

Théophile Chaumont-Frelet, Diego Paredes, and Frédéric Valentin (2026). Flux approximation on unfitted meshes and application to multiscale hybrid-mixed methods, Computers & Mathematics with Applications 209, 16–27. DOI: 10.1016/j.camwa.2026.01.016.

Harder, Paredes and Valentin (2015): Advective/reactive domination

Christopher Harder, Diego Paredes, and Frédéric Valentin (2015). On a Multiscale Hybrid-Mixed Method for Advective-Reactive Dominated Problems with Heterogeneous Coefficients, Multiscale Modeling & Simulation 13(2), 491–518. DOI: 10.1137/130938499.

Araya et al. (2024): Generalized RAD on polytopes

Rodolfo Araya, Fabrice Jaillet, Diego Paredes, and Frédéric Valentin (2024). Generalizing the multiscale hybrid-mixed method for reactive-advective-diffusive equations, Computer Methods in Applied Mechanics and Engineering 428, 117089. DOI: 10.1016/j.cma.2024.117089.

Araya et al. (2017): Stokes and Brinkman construction

Rodolfo Araya, Christopher Harder, Abner H. Poza, and Frédéric Valentin (2017). Multiscale hybrid-mixed method for the Stokes and Brinkman equations—The method, Computer Methods in Applied Mechanics and Engineering 324, 29–53. DOI: 10.1016/j.cma.2017.05.027.

Araya, Rebolledo and Valentin (2021): Multilevel Stokes/Brinkman estimator

Rodolfo Araya, Ramiro Rebolledo, and Frédéric Valentin (2021). On a multiscale a posteriori error estimator for the Stokes and Brinkman equations, IMA Journal of Numerical Analysis 41(1), 344–380. DOI: 10.1093/imanum/drz053. An earlier version is HAL: hal-01945934v1.

Araya et al. (2021): Adaptive Oseen

Rodolfo Araya, Cristian Cárcamo, Abner H. Poza, and Frédéric Valentin (2021). An adaptive multiscale hybrid-mixed method for the Oseen equations, Advances in Computational Mathematics 47, article 15. DOI: 10.1007/s10444-020-09833-8.

Araya et al. (2025): Stokes/Brinkman a priori analysis

Rodolfo Araya, Christopher Harder, Abner H. Poza, and Frédéric Valentin (2025). Multiscale Hybrid-Mixed Methods for the Stokes and Brinkman Equations—A Priori Analysis, SIAM Journal on Numerical Analysis 63(2), 588–618. DOI: 10.1137/24M1649368.

Harder, Madureira and Valentin (2016): Primal hybrid elasticity

Christopher Harder, Alexandre L. Madureira, and Frédéric Valentin (2016). A hybrid-mixed method for elasticity, ESAIM: M2AN 50, 311–336. DOI: 10.1051/m2an/2015046.

Devloo et al. (2021): Mixed elasticity with weak stress symmetry

Philippe R. B. Devloo, Agnaldo M. Farias, Sônia M. Gomes, Weslley Pereira, Antonio J. B. dos Santos, and Frédéric Valentin (2021). New H(div)-conforming multiscale hybrid-mixed methods for the elasticity problem on polygonal meshes, ESAIM: M2AN 55, 1005–1037. DOI: 10.1051/m2an/2021013.

Gomes, Pereira and Valentin (2024, preprint v1): Low-order locking-free elasticity

Antônio Tadeu Azevedo Gomes, Weslley da Silva Pereira, and Frédéric Valentin (2024). A low-order locking-free multiscale finite element method for isotropic elasticity, arXiv preprint, version 1, 25 March 2024. arXiv: 2403.16890v1.

Gomes et al. (2017, preprint v1): Scalable implementation

Antônio Tadeu A. Gomes, Weslley S. Pereira, Frédéric Valentin, and Diego Paredes (2017). On the Implementation of a Scalable Simulator for Multiscale Hybrid-Mixed Methods, arXiv preprint, version 1, 30 March 2017. arXiv: 1703.10435v1.

Other implemented multiscale methods

  • Stéphane Lanteri, Diego Paredes, Claire Scheid, and Frédéric Valentin (2018). The Multiscale Hybrid-Mixed method for the Maxwell Equations in Heterogeneous Media. Multiscale Modeling & Simulation 16(4) 1648-1683. DOI: 10.1137/16M110037X.

  • Théophile Chaumont-Frelet, and Frédéric Valentin (2020). A Multiscale Hybrid-Mixed Method for the Helmholtz Equation in Heterogeneous Domains. SIAM Journal on Numerical Analysis 58(2) 1029-1067. DOI: 10.1137/19M1255616.

  • Antonio Tadeu Gomes, Diego Paredes, Weslley Pereira, Roberto Souto, and Frederic Valentin (2017). A Multiscale Hybrid-Mixed Method for the Elastodynamic Model with Rough Coefficients. Proceedings of the XXXVIII Iberian Latin American Congress on Computational Methods in Engineering. DOI: 10.20906/CPS/CILAMCE2017-0399.

  • Honório Fernando, Larissa Martins, Weslley Pereira, and Frédéric Valentin (2023). A Petrov–Galerkin multiscale hybrid-mixed method for the Darcy equation on polytopes. Computational and Applied Mathematics 42, article 173. DOI: 10.1007/s40314-023-02304-y.

  • Gabriel R. Barrenechea, Antonio Tadeu A. Gomes, and Diego Paredes (2024). A Multiscale Hybrid Method. SIAM Journal on Scientific Computing 46(3), A1628–A1657. DOI: 10.1137/22M1542556.

  • Larissa Martins, Juan F. Pacazuca, and Frédéric Valentin (2025). A Multiscale Hybrid-Mixed Method with Local Stabilization. Proceedings of the Ibero-Latin American Congress on Computational Methods in Engineering, CILAMCE 2025, volume 5, article 14270; published online 18 March 2026. DOI: 10.55592/cilamce2025.v5i.14270.

  • Franklin de Barros, Alexandre L. Madureira, and Frédéric Valentin (2026). A three-field Multiscale Method. arXiv preprint, version 3, 5 August 2026; first submitted 25 April 2024. arXiv: 2404.16978v3.

  • Douglas N. Arnold, Richard S. Falk, and Ragnar Winther (2007). Mixed finite element methods for linear elasticity with weakly imposed symmetry. Mathematics of Computation 76, 1699–1723. DOI: 10.1090/S0025-5718-07-01998-9. Preprint: arXiv:math/0701506v1.

Finite elements, stabilization, meshing, software and data

  • M. Cecilia Rivara (1984). Algorithms for refining triangular grids suitable for adaptive and multigrid techniques. International Journal for Numerical Methods in Engineering 20(4), 745–756. DOI: 10.1002/nme.1620200412.

  • Pedro Henrique Penna, Antônio Tadeu A. Gomes, Márcio Castro, Patricia D.M. Plentz, Henrique C. Freitas, François Broquedis, and Jean‐François Méhaut (2019). A comprehensive performance evaluation of the BinLPT workload‐aware loop scheduler. Concurrency and Computation: Practice and Experience 31(18) e5170. DOI: 10.1002/cpe.5170.

  • Gary S. Martin, Robert Wiley, and Kurt J. Marfurt (2006). Marmousi2: An elastic upgrade for Marmousi. The Leading Edge 25(2) 156-166. DOI: 10.1190/1.2172306.

  • Douglas A. Castro, Philippe R.B. Devloo, Agnaldo M. Farias, Sônia M. Gomes, Denise de Siqueira, and Omar Durán (2016). Three dimensional hierarchical mixed finite element approximations with enhanced primal variable accuracy. Computer Methods in Applied Mechanics and Engineering 306 479-502. DOI: 10.1016/j.cma.2016.03.050.

  • Gabriel R. Barrenechea, Fabrice Jaillet, Diego Paredes, and Frédéric Valentin (2020). The multiscale hybrid mixed method in general polygonal meshes. Numerische Mathematik 145(1), 197–237. DOI: 10.1007/s00211-020-01103-5.

  • L. P. Franca and Frédéric Valentin (2000). On an improved unusual stabilized finite element method for the advective–reactive–diffusive equation. Computer Methods in Applied Mechanics and Engineering 190(13–14), 1785–1800. DOI: 10.1016/S0045-7825(00)00190-0.

  • Martin S. Alnæs, Anders Logg, Kristian B. Ølgaard, Marie E. Rognes, and Garth N. Wells (2014). Unified Form Language: A domain-specific language for weak formulations of partial differential equations. ACM Transactions on Mathematical Software 40(2), article 9, 1–37. DOI: 10.1145/2566630. Author preprint: arXiv:1211.4047v2.

  • M. A. Christie and M. J. Blunt (2001). Tenth SPE Comparative Solution Project: A Comparison of Upscaling Techniques. SPE Reservoir Evaluation & Engineering 4(4), 308–317. SPE10 dataset and benchmark description.

  • Basix documentation: reference finite elements, basis tabulation, degree-of-freedom transformations and mappings.

  • DOLFINx documentation: native finite element assembly and distributed mesh/function interfaces.
  • PETSc documentation: distributed matrices, solvers and preconditioners.
  • Gmsh documentation: geometry, mesh generation and physical groups.

  • meshio project: supported mesh exchange formats and cell/point metadata.

  • Willy Dörfler (1996). A Convergent Adaptive Algorithm for Poisson’s Equation. SIAM Journal on Numerical Analysis 33(3), 1106–1124. DOI.

Software provenance

Project or source Revision or documentation
MSL: msl_mhm 4cb8cf81518284313b680b13fd586ee619f08b99
MSL: msl_cg afb76d14c1baf50f0b9e69f7bcac675749ef4458
MSL: msl_core 7f15f455717173d29080d411a7e732c72c1e87f8
msl_mfem b9a67e7079c7e487e4ab1679c1bb3c880cc1909a; MFEM 4.9
mhm-mfem fb535acec1be87c19b3aa538ac73a265570e9fdc; MFEM 4.9
DOLFINx/UFL Documentation; DOLFINx 0.9.0
NeoPZ Revision 4c6b6d277ce097b97bfc8dea1b6725860f4fe05a
Labmec/MHM Revision f978f29d657d28fe58bcea20fabee68953093482

The MSL repositories are not public. Their source code and comparison drivers are not distributed with PyMHM.