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Mathematics of Computation

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Second order splitting for a class of fourth order equations


Authors: Charles M. Elliott, Hans Fritz and Graham Hobbs
Journal: Math. Comp. 88 (2019), 2605-2634
MSC (2010): Primary 65N30, 65J10, 35J35
DOI: https://doi.org/10.1090/mcom/3425
Published electronically: April 1, 2019
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Abstract: We formulate a well-posedness and approximation theory for a class of generalised saddle point problems. In this way we develop an approach to a class of fourth order elliptic partial differential equations using the idea of splitting into coupled second order equations. Our main motivation is to treat certain fourth order equations on closed surfaces arising in the modelling of biomembranes but the approach may be applied more generally. In particular we are interested in equations with non-smooth right-hand sides and operators which have non-trivial kernels. 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
Email: C.M.Elliott@warwick.ac.uk

Hans Fritz
Affiliation: Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany
Email: fritz.hans@web.de

Graham Hobbs
Affiliation: Mathematics Institute, Zeeman Building, University of Warwick, Coventry, CV4 7AL, United Kingdom
Email: graham.hobbs@warwickgrad.net

DOI: https://doi.org/10.1090/mcom/3425
Received by editor(s): November 11, 2017
Received by editor(s) in revised form: October 5, 2018, and December 16, 2018
Published electronically: April 1, 2019
Additional Notes: The work of the first author was partially supported by the Royal Society via a Wolfson Research Merit Award.
The second author thanks the Alexander von Humboldt Foundation, Germany, for their financial support by a Feodor Lynen Research Fellowship in collaboration with the University of Warwick, UK
The research of the third 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 2019 American Mathematical Society