FEniCSx-pctools: Tools for PETSc Block Linear Algebra Preconditioning in FEniCSx
By: Martin Řehoř and Jack S. Hale
References
- Allaire G, Kaber SM.
Numerical linear algebra . In: Marsden, JE, Sirovich, L, Antman, SS, editors. Texts in Applied Mathematics, vol. 55. New York, NY: Springer; 2008. ISBN: 978-0-387-34159-0 978-0-387-68918-0. DOI: 10.1007/978-0-387-68918-0 - Alnæs M, et al. The FEniCS Project Version 1.5. Archive of Numerical Software. Dec. 2015;3(100). ISSN: 2197-8263. DOI: 10.11588/ans.2015.100.20553
- Alnæs MS, et al. Unified Form Language: A Domain-specific Language for Weak Formulations of Partial Differential Equations. ACM Trans Math Softw. Mar. 2014;40(2):9:1–9:37. ISSN: 0098-3500. DOI: 10.1145/2566630
- Baker CG, Heroux MA. Tpetra, and the use of generic programming in scientific computing. Sci Program. Apr 2012;20(2):115–128. DOI: 10.1155/2012/693861
- Balay S, et al. PETSc/TAO Users Manual. Tech. rep. ANL-21/39 – Revision 3.20. Argonne National Laboratory; 2023. DOI: 10.2172/2205494
- Baratta IA, et al. DOLFINx: The next generation FEniCS problem solving environment. Dec. 2023. DOI: 10.5281/zenodo.10447666
- Benzi M, Golub GH, Liesen J. Numerical solution of saddle point problems. Acta Numerica. May 2005;14:1–137. Cambridge University Press. ISSN: 1474-0508, 0962-4929. DOI: 10.1017/S0962492904000212
- Blechta J. Towards efficient numerical computation of flows of non-Newtonian fluids. PhD thesis. Univerzita Karlova; June 2019. URL:
https://dspace.cuni.cz/handle/20.500.11956/108384 - Blechta J, Řehoř M. FENaPack – FEniCS Navier–Stokes preconditioning package. In: Hale, JS, editor. Proceedings of the FEniCS Conference 2017. Luxembourg:
University of Luxembourg ;June 2017 . pp. 14–15. DOI: 10.6084/m9.figshare.5086369 - Brown J, et al.
Composable Linear Solvers for Multiphysics . In: 2012 11th International Symposium on Parallel and Distributed Computing (ISPDC). Munich, Germany: IEEE; June 2012. pp. 55–62. ISBN: 978-1-4673-2599-8. URL:http://ieeexplore.ieee.org/document/6341494/ - Budiša A, et al. HAZniCS – Software Components for Multiphysics Problems. ACM Transactions on Mathematical Software. Dec. 31, 2023;49(4):1–23. ISSN: 0098-3500, 1557-7295. URL:
https://dl.acm.org/doi/10.1145/3625561 - Budiša A, et al.
Rational Approximation Preconditioners for Multiphysics Problems . In: Georgiev, I, et al. editors. Numerical Methods and Applications. Lecture Notes in Computer Science. Cham: Springer Nature Switzerland; 2023. 100–113. ISBN: 978-3-031-32412-3. DOI: 10.1007/978-3-031-32412-3_9 - Chang J, Protasov K, Allen JM. PFIBS: a Parallel FEniCS Implementation of Block Solvers; 2022. URL:
https://github.com/NREL/pfibs - Chen S, et al.
Robust block preconditioners for poroelasticity . In: Computer Methods in Applied Mechanics and Engineering. Sept. 2020;369:113229 . ISSN: 0045-7825. DOI: 10.1016/j.cma.2020.113229 - Dokken JS. The FEniCSx tutorial. URL:
https://jsdokken.com/dolfinx-tutorial/ - Elman H, Silvester D, Wathen A. Finite Elements and Fast Iterative Solvers: with Applications in Incompressible Fluid Dynamics. Oxford University Press; June 2014. ISBN: 978-0-19-967879-2. DOI: 10.1093/acprof:oso/9780199678792.001.0001
- Farrell PE, et al.
A Reynolds-robust preconditioner for the Scott–Vogelius discretization of the stationary incompressible Navier–Stokes equations . In: The SMAI Journal of Computational Mathematics. 2021;7:75–96. ISSN: 2426-8399. DOI: 10.5802/smai-jcm.72 - Kirby RC, Mitchell L. Solver Composition Across the PDE/Linear Algebra Barrier. SIAM Journal on Scientific Computing. Jan. 2018;40(1):C76–C98. ISSN: 1064-8275, 1095-7197. DOI: 10.1137/17M1133208
- Laakmann F, Farrell PE, Mitchell L.
An Augmented Lagrangian Preconditioner for the Magnetohydrodynamics Equations at High Reynolds and Coupling Numbers . SIAM Journal on Scientific Computing. Aug. 2022;44(4).B1018–B1044. Society for Industrial and Applied Mathematics. ISSN: 1064-8275. DOI: 10.1137/21M1416539 - Lange M, et al.
Efficient Mesh Management in Firedrake Using PETSc DMPlex . SIAM Journal on Scientific Computing. Jan. 2016;38(5):S143–S155. Society for Industrial and Applied Mathematics. ISSN: 1064-8275. DOI: 10.1137/15M1026092 - Anders L, Wells GN. DOLFIN: Automated Finite Element Computing. ACM Trans Math Softw. Apr. 2010;37(2):20:1–20:28. ISSN: 0098-3500. DOI: 10.1145/1731022.1731030
- Kent-Andre M, Joachim BH.
Block preconditioning of sys-tems of PDEs . In: Logg, A, Mardal, K-A, Wells, G, editors. Automated Solution of Differential Equations by the Finite Element Method. Lecture Notes in Computational Science and Engineering 84. Berlin Heidelberg: Springer; Jan. 2012. pp. 643–655. ISBN: 978-3-642-23098-1 978-3-642-23099-8. DOI: 10.1007/978-3-642-23099-8_35 - Rathgeber F, et al. Firedrake: Automating the Finite Element Method by Composing Abstractions. ACM Transactions on Mathematical Software (TOMS). Dec. 2016;43(3):24:1–24:27. ISSN: 0098-3500. DOI: 10.1145/2998441
- Rusten T, Vassilevski P, Winther R. Interior penalty preconditioners for mixed finite element approximations of elliptic problems. Mathematics of Computation. 1996;65(214):447–466. ISSN: 0025-5718, 1088-6842. DOI: 10.1090/S0025-5718-96-00720-X
- The Trilinos Project Team. The Trilinos Project Website. URL:
https://trilinos.github.io - Varrette S, et al. Management of an Academic HPC & Research Computing Facility: The ULHPC Experience 2.0. In: Proceedings of the 2022 6th High Performance Computing and Cluster Technologies Conference. HPCCT’22. New York, NY, USA:
Association for Computing Machinery ;Oct. 2022 . pp. 14–24. DOI: 10.1145/3560442.3560445 - Řehoř M, Hale JS. FEniCSx Preconditioning Tools. Sept. 2025. DOI: 10.6084/m9.figshare.21408294.v6
DOI: https://doi.org/10.5334/jors.494 | Journal eISSN: 2049-9647
Language: English
Submitted on: Nov 24, 2023
Accepted on: Sep 5, 2025
Published on: Sep 22, 2025
Published by: Ubiquity Press
In partnership with: Paradigm Publishing Services
Publication frequency: 1 issue per year
Keywords:
© 2025 Martin Řehoř, Jack S. Hale, published by Ubiquity Press
This work is licensed under the Creative Commons Attribution 4.0 License.