Geodesic
A unified analysis of three finite element methods for the Monge-Ampère equation
Electronic transactions on numerical analysis, Tome 41 (2014), pp. 262-288
It was recently shown in S. C. Brenner et al. [Math. Comp., 80 (2011), pp. 1979 -- 1995] that Lagrange finite elements can be used to approximate classical solutions of the Monge-Ampère equation, a fully nonlinear second order PDE. We expand on these results and give a unified analysis for many finite element methods satisfying some mild structure conditions in two and three dimensions. After proving some abstract results, we lay out a blueprint to construct various finite element methods that inherit these conditions and show how $C^1$ finite element methods, $C^0$ finite element methods, and discontinuous Galerkin methods fit into the framework.
Classification : 65N30, 65N12, 35J60
Keywords: fully nonlinear pdes, Monge-Ampère equation, finite element methods, discontinuous Galerkin methods 
@article{ETNA_2014__41__a11,
     author = {Neilan,  Michael},
     title = {A unified analysis of three finite element methods for the {Monge-Amp\`ere} equation},
     journal = {Electronic transactions on numerical analysis},
     pages = {262--288},
     year = {2014},
     volume = {41},
     zbl = {1302.65256},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ETNA_2014__41__a11/}
}
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Neilan,  Michael. A unified analysis of three finite element methods for the Monge-Ampère equation. Electronic transactions on numerical analysis, Tome 41 (2014), pp. 262-288. http://geodesic.mathdoc.fr/item/ETNA_2014__41__a11/