Volume 51, Number 2, March-April 2017
|Page(s)||707 - 725|
|Published online||10 March 2017|
Standard finite elements for the numerical resolution of the elliptic Monge–Ampère equation: Aleksandrov solutions
Department of Mathematics, Statistics, and Computer Science, M/C 249. University of Illinois at Chicago, Chicago, IL 60607-7045, USA
Received: 31 July 2015
Revised: 10 February 2016
Accepted: 16 May 2016
We prove a convergence result for a natural discretization of the Dirichlet problem of the elliptic Monge–Ampère equation using finite dimensional spaces of piecewise polynomial C1 functions. Discretizations of the type considered in this paper have been previously analyzed in the case the equation has a smooth solution and numerous numerical evidence of convergence were given in the case of non smooth solutions. Our convergence result is valid for non smooth solutions, is given in the setting of Aleksandrov solutions, and consists in discretizing the equation in a subdomain with the boundary data used as an approximation of the solution in the remaining part of the domain. Our result gives a theoretical validation for the use of a non monotone finite element method for the Monge–Ampère equation.
Mathematics Subject Classification: 35J96 / 65N30
Key words: Weak convergence / Monge–Ampère measure / Aleksandrov solution / finite elements
© EDP Sciences, SMAI 2017
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