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 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.
Accepté le :
DOI : 10.1051/m2an/2016037
Mots clés : Weak convergence, Monge–Ampère measure, Aleksandrov solution, finite elements
@article{M2AN_2017__51_2_707_0, author = {Awanou, Gerard}, title = {Standard finite elements for the numerical resolution of the elliptic {Monge{\textendash}Amp\`ere} equation: {Aleksandrov} solutions}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis }, pages = {707--725}, publisher = {EDP-Sciences}, volume = {51}, number = {2}, year = {2017}, doi = {10.1051/m2an/2016037}, mrnumber = {3626416}, zbl = {1372.35120}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an/2016037/} }
TY - JOUR AU - Awanou, Gerard TI - Standard finite elements for the numerical resolution of the elliptic Monge–Ampère equation: Aleksandrov solutions JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2017 SP - 707 EP - 725 VL - 51 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2016037/ DO - 10.1051/m2an/2016037 LA - en ID - M2AN_2017__51_2_707_0 ER -
%0 Journal Article %A Awanou, Gerard %T Standard finite elements for the numerical resolution of the elliptic Monge–Ampère equation: Aleksandrov solutions %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2017 %P 707-725 %V 51 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2016037/ %R 10.1051/m2an/2016037 %G en %F M2AN_2017__51_2_707_0
Awanou, Gerard. Standard finite elements for the numerical resolution of the elliptic Monge–Ampère equation: Aleksandrov solutions. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 2, pp. 707-725. doi : 10.1051/m2an/2016037. http://www.numdam.org/articles/10.1051/m2an/2016037/
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