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A unified theory for continuous-in-time evolving finite element space approximations to partial differential equations in evolving domains
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Elliott, Charles M. and Ranner, Thomas (2021) A unified theory for continuous-in-time evolving finite element space approximations to partial differential equations in evolving domains. IMA Journal of Numerical Analysis, 41 (3). pp. 1696-1845. draa062. doi:10.1093/imanum/draa062 ISSN 0272-4979.
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WRAP-unified-theory-continuous-time-evolving-finite-element-space-approximations-partial-differential-equations-evolving-domains-Elliott-2020.pdf - Accepted Version - Requires a PDF viewer. Download (1856Kb) | Preview |
Official URL: https://doi.org/10.1093/imanum/draa062
Abstract
We develop a unified theory for continuous-in-time finite element discretizations of partial differential equations posed in evolving domains, including the consideration of equations posed on evolving surfaces and bulk domains, as well as coupled surface bulk systems. We use an abstract variational setting with time-dependent function spaces and abstract time-dependent finite element spaces. Optimal a priori bounds are shown under usual assumptions on perturbations of bilinear forms and approximation properties of the abstract finite element spaces. The abstract theory is applied to evolving finite elements in both flat and curved spaces. Evolving bulk and surface isoparametric finite element spaces defined on evolving triangulations are defined and developed. These spaces are used to define approximations to parabolic equations in general domains for which the abstract theory is shown to apply. Numerical experiments are described, which confirm the rates of convergence.
| Item Type: | Journal Article | ||||||||
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| Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||
| Journal or Publication Title: | IMA Journal of Numerical Analysis | ||||||||
| Publisher: | Oxford University Press | ||||||||
| ISSN: | 0272-4979 | ||||||||
| Official Date: | July 2021 | ||||||||
| Dates: |
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| Volume: | 41 | ||||||||
| Number: | 3 | ||||||||
| Page Range: | pp. 1696-1845 | ||||||||
| Article Number: | draa062 | ||||||||
| DOI: | 10.1093/imanum/draa062 | ||||||||
| Status: | Peer Reviewed | ||||||||
| Publication Status: | Published | ||||||||
| Reuse Statement (publisher, data, author rights): | This is a pre-copyedited, author-produced version of an article accepted for publication in IMA Journal of Numerical Analysis following peer review. The version of record C M Elliott, T Ranner, A unified theory for continuous-in-time evolving finite element space approximations to partial differential equations in evolving domains, IMA Journal of Numerical Analysis, Volume 41, Issue 3, July 2021, Pages 1696–1845, https://doi.org/10.1093/imanum/draa062, is available online at: https://doi.org/10.1093/imanum/draa062 | ||||||||
| Access rights to Published version: | Restricted or Subscription Access | ||||||||
| Date of first compliant deposit: | 13 August 2020 | ||||||||
| Date of first compliant Open Access: | 26 November 2021 | ||||||||
| Is Part Of: | 1 | ||||||||
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