DOLFIN: Automated finite element computing
Open accessAbstract
We describe here a library aimed at automating the solution of partial differential equations using the finite element method. By employing novel techniques for automated code generation, the library combines a high level of expressiveness with efficient computation. Finite element variational forms may be expressed in near mathematical notation, from which low-level code is automatically generated, compiled, and seamlessly integrated with efficient implementations of computational meshes and high-performance linear algebra. Easy-to-use object-oriented interfaces to the library are provided in the form of a C++ library and a Python module. This article discusses the mathematical abstractions and methods used in the design of the library and its implementation. A number of examples are presented to demonstrate the use of the library in application code.
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The treatment of cell migration via continuum PDEs is well established in biophysics, as is the role of cell arrangement in determining aspects of the progression of cancers.
“The treatment of cell migration via continuum PDEs is well established in biophysics36,37,38, as is the role of cell arrangement in determining aspects of the progression of cancers39.”
From Inference of weak-form partial differential equations describing migration and proliferation mechanisms in wound healing experiments on cancer cells (2025) · cites this paper as [38]
The citation is attached to the first clause, about continuum-PDE treatment of cell migration; citation 39 is attached separately to the cancer claim. The Introduction describes PDEs generally, and §5.3 gives a convection–diffusion example, but neither addresses cell migration or its established use in biophysics. Searches for cell migration and biophysics found no relevant passage in our copy. (LLM, conf=0.76)
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Research works printed but not linked: 17
- Bangerth, W., Hartmann, R., and Kanschat, G. 2007. deal.II -A general purpose object oriented finite element library. ACM Transactions on Mathematical Software 33, 4, 24.
- Berti, G. 2002. Generic programming for mesh algorithms: Towards universally usable geo- metric components. In Proceedings of the Fifth World Congress on Computational Mechanics (WCCM V), H. A. Mang, F. G. Rammerstorfer, and J. Eberhardsteiner, Eds. Vienna University of Technology, Vienna. http://wccm.tuwien.ac.at/publications/Papers/fp81327.pdf.
- Brezzi, F. and Fortin, M. 1991. Mixed and Hybrid Finite Element Methods. Springer Series in Computational Mathematics, vol. 15. Springer, New York.
- Chen, Y., Davis, T. A., Hager, W. W., and Rajamanickam, S. 2008. Algorithm 887: CHOLMOD, supernodal sparse Cholesky factorization and update/downdate. ACM Trans- actions on Mathematical Software 35, 3, 1-14.
- Davis, T. A. 2004. Algorithm 832: UMFPACK v4.3-an unsymmetric-pattern multifrontal method. ACM Transactions on Mathematical Software 30, 2, 196-199.
- Gamma, E., Helm, R., Johnson, R., and Vlissides, J. 1995. Design Patterns: Elements of Reusable Object-Oriented Software. Addison Wesley.
- Heroux, M. A., Bartlett, R. A., Howle, V. E., Hoekstra, R. J., Hu, J. J., Kolda, T. G., Lehoucq, R. B., Long, K. R., Pawlowski, R. P., Phipps, E. T., Salinger, A. G., Thorn- quist, H. K., Tuminaro, R. S., Willenbring, J. M., Williams, A., and Stanley, K. S. 2005. An overview of the Trilinos project. ACM Transactions on Mathematical Software 31, 3, 397-423.
- Kirby, R. C. 2004. Algorithm 839: FIAT, a new paradigm for computing finite element basis functions. ACM Transactions on Mathematical Software 30, 4 (Dec.), 502-516.
- Kirby, R. C. 2006. Optimizing FIAT with Level 3 BLAS. ACM Transactions on Mathematical Software 32, 2 (June), 223-235.
- Kirby, R. C., Knepley, M. G., Logg, A., and Scott, L. R. 2005. Optimizing the evaluation of finite element matrices. SIAM Journal on Scientific Computing 27, 3, 741-758.
- Kirby, R. C. and Logg, A. 2006. A compiler for variational forms. ACM Transactions on Mathematical Software 32, 3, 417-444.
- Kirby, R. C. and Logg, A. 2007. Efficient compilation of a class of variational forms. ACM Transactions on Mathematical Software 33, 3.
- Kirby, R. C., Logg, A., Scott, L. R., and Terrel, A. R. 2006. Topological optimization of the evaluation of finite element matrices. SIAM Journal on Scientific Computing 28, 1, 224-240.
- Langtangen, H. P. 2003. Computational Partial Differential Equations: Numerical Methods and Diffpack Programming. Texts in Computational Science and Engineering, vol. 1. Springer.
- Logg, A. 2009. Efficient representation of computational meshes. International Journal of Computational Science and Engineering. To appear.
- Ølgaard, K. B., Logg, A., and Wells, G. N. 2008. Automated code generation for discontin- uous Galerkin methods. SIAM Journal on Scientific Computing 31, 2, 849-864.
- Ølgaard, K. B., Wells, G. N., and Logg, A. 2008. Automated computational modelling for solid mechanics. In IUTAM Symposium on Theoretical, Computational and Modelling Aspects of Inelastic Media, B. D. Reddy, Ed. IUTAM Bookseries, vol. 11. Springer, 192-204.
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Srivastava, S., Kinnunen, P. C., Wang, Z., Ho, K. K. Y., Humphries, B. A., Chen, S., Linderman, J. J., Luker, G. D., Luker, K. E., & Garikipati, K. (2025) · PLOS Computational Biology · doi:10.1371/journal.pcbi.1013607 · Cited by 5 · see citing works
“The numerical examples presented in this work have been posed and solved in two dimensions by the finite element method implemented on the FEniCS platform [38,39].”
- Authors
- Logg, A., & Wells, G. N.
- Affiliations
- University of Oslo (NO)
- University of Cambridge (GB)
- Venue
- ACM Transactions on Mathematical Software
- Published
- 2010-04
- Volume
- 37
- Issue
- 2
- Pages
- 1-28
- ISSN
- 0098-3500 (ISSN-L)
- 1557-7295
- OpenAlex
- W2128095813
- Cited by
- 750
- References
- 0
- Citations by year
- 2026: 24 · 2025: 57 · 2024: 73 · 2023: 76 · 2022: 81 · 2021: 56 · 2020: 54 · 2019: 40 · 2018: 42 · 2017: 44 · 2016: 41 · 2015: 32 · 2014: 30 · 2013: 34 · 2012: 31
- FWCI
- 30.84
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- ACM Transactions on Mathematical Software (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- Apollo (University of Cambridge) (submittedVersion)
- Cambridge University Engineering Department Publications Database (submittedVersion)
- Chalmers Publication Library (Chalmers University of Technology) (submittedVersion)
- Chalmers Research (Chalmers University of Technology) (submittedVersion)
- arXiv (Cornell University)
- Funders
- Norges Forskningsråd (NFR 180450)
- Topics
- Physical Sciences › Engineering › Computational Mechanics › Advanced Numerical Methods in Computational Mathematics
- Physical Sciences › Computer Science › Computational Theory and Mathematics › Matrix Theory and Algorithms
- Physical Sciences › Mathematics › Modeling and Simulation › Mathematical and Computational Methods
- Keywords
- finite element analysis, partial differential equations, automatic code generation, variational formulation, computational mesh, object-oriented design, Python module, C++ library
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