In this paper, we construct and analyze finite element methods for the three dimensional Monge-Ampère equation. We derive methods using the Lagrange finite element space such that the resulting discrete linearizations are symmetric and stable. With this in hand, we then prove the well-posedness of the method, as well as derive quasi-optimal error estimates. We also present some numerical experiments that back up the theoretical findings.
@article{M2AN_2012__46_5_979_0,
author = {Brenner, Susanne Cecelia and Neilan, Michael},
title = {Finite element approximations of the three dimensional Monge-Amp\`ere equation},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
volume = {46},
year = {2012},
pages = {979-1001},
doi = {10.1051/m2an/2011067},
mrnumber = {2916369},
zbl = {1272.65088},
language = {en},
url = {http://dml.mathdoc.fr/item/M2AN_2012__46_5_979_0}
}
Brenner, Susanne Cecelia; Neilan, Michael. Finite element approximations of the three dimensional Monge-Ampère equation. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 46 (2012) pp. 979-1001. doi : 10.1051/m2an/2011067. http://gdmltest.u-ga.fr/item/M2AN_2012__46_5_979_0/
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