## A unified analysis of three finite element methods for the Monge-Ampère equation

Michael Neilan

### Abstract

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.

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### Key words

fully nonlinear PDEs, Monge-Ampère equation, finite element methods, discontinuous Galerkin methods

### AMS subject classifications

65N30, 65N12, 35J60.

### Links to the cited ETNA articles

 [9] Vol. 18 (2004), pp. 42-48 Susanne C. Brenner: Discrete Sobolev and Poincaré inequalities for piecewise polynomial functions

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