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Code generation for generally mapped finite elements

Kirby, Robert C; Mitchell, Lawrence

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Authors

Robert C Kirby

Lawrence Mitchell



Abstract

Many classical finite elements such as the Argyris and Bell elements have long been absent from high-level PDE software. Building on recent theoretical work, we describe how to implement very general finite-element transformations in FInAT and hence into the Firedrake finite-element system. Numerical results evaluate the new elements, comparing them to existing methods for classical problems. For a second-order model problem, we find that new elements give smooth solutions at a mild increase in cost over standard Lagrange elements. For fourth-order problems, however, the newly enabled methods significantly outperform interior penalty formulations. We also give some advanced use cases, solving the nonlinear Cahn-Hilliard equation and some biharmonic eigenvalue problems (including Chladni plates) using C1 discretizations.

Citation

Kirby, R. C., & Mitchell, L. (2019). Code generation for generally mapped finite elements. ACM Transactions on Mathematical Software, 45(4), Article 41. https://doi.org/10.1145/3361745

Journal Article Type Article
Acceptance Date Sep 9, 2019
Publication Date Dec 31, 2019
Deposit Date Aug 29, 2018
Publicly Available Date Jan 21, 2020
Journal ACM Transactions on Mathematical Software
Print ISSN 0098-3500
Electronic ISSN 1557-7295
Publisher Association for Computing Machinery (ACM)
Peer Reviewed Peer Reviewed
Volume 45
Issue 4
Article Number 41
DOI https://doi.org/10.1145/3361745
Related Public URLs https://arxiv.org/pdf/1808.05513.pdf

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Copyright Statement
© 2019 Copyright held by the owner/author(s). This is the author's version of the work. It is posted here for your personal use. Not for redistribution. The definitive Version of Record was published in ACM transactions on mathematical software, https://doi.org/10.1145/3361745




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