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Mathematics of Computation

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Second order splitting of a class of fourth order PDEs with point constraints


Authors: Charles M. Elliott and Philip J. Herbert
Journal: Math. Comp. 89 (2020), 2613-2648
MSC (2010): Primary 65N30, 65J10, 35J35
DOI: https://doi.org/10.1090/mcom/3556
Published electronically: July 27, 2020
MathSciNet review: 4136541
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Abstract: We formulate a well-posedness and approximation theory for a class of generalised saddle point problems with a specific form of constraints. In this way we develop an approach to a class of fourth order elliptic partial differential equations with point constraints using the idea of splitting into coupled second order equations. An approach is formulated using a penalty method to impose the constraints. Our main motivation is to treat certain fourth order equations involving the biharmonic operator and point Dirichlet constraints for example arising in the modelling of biomembranes on curved and flat surfaces but the approach may be applied more generally. The theory for well-posedness and approximation is presented in an abstract setting. Several examples are described together with some numerical experiments.


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Additional Information

Charles M. Elliott
Affiliation: Mathematics Institute, Zeeman Building, University of Warwick, Coventry, CV4 7AL, United Kingdom
MR Author ID: 62960
ORCID: 0000-0002-6924-4455
Email: C.M.Elliott@warwick.ac.uk

Philip J. Herbert
Affiliation: Mathematics Institute, Zeeman Building, University of Warwick, Coventry, CV4 7AL, United Kingdom
ORCID: 0000-0002-6513-1728
Email: P.J.Herbert@warwick.ac.uk

Received by editor(s): November 7, 2019
Received by editor(s) in revised form: April 2, 2020
Published electronically: July 27, 2020
Additional Notes: The work of the first author was partially supported by the Royal Society via a Wolfson Research Merit Award.
The research of the second author was funded by the Engineering and Physical Sciences Research Council grant EP/H023364/1 under the MASDOC centre for doctoral training at the University of Warwick.
Article copyright: © Copyright 2020 American Mathematical Society