We address in this article the computation of the convex solutions of the Dirichlet problem for the real elliptic Monge - Ampère equation for general convex domains in two dimensions. The method we discuss combines a least-squares formulation with a relaxation method. This approach leads to a sequence of Poisson - Dirichlet problems and another sequence of low dimensional algebraic eigenvalue problems of a new type. Mixed finite element approximations with a smoothing procedure are used for the computer implementation of our least-squares/relaxation methodology. Domains with curved boundaries are easily accommodated. Numerical experiments show the convergence of the computed solutions to their continuous counterparts when such solutions exist. On the other hand, when classical solutions do not exist, our methodology produces solutions in a least-squares sense.

Keywords: Monge − ampère equation, least-squares method, biharmonic problem, conjugate gradient method, quadratic constraint minimization, mixed finite element methods

@article{COCV_2013__19_3_780_0, author = {Caboussat, Alexandre and Glowinski, Roland and Sorensen, Danny C.}, title = {A least-squares method for the numerical solution of the {Dirichlet} problem for the elliptic monge-amp\`ere equation in dimension two}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {780--810}, publisher = {EDP-Sciences}, volume = {19}, number = {3}, year = {2013}, doi = {10.1051/cocv/2012033}, zbl = {1272.65089}, mrnumber = {3092362}, language = {en}, url = {http://www.numdam.org/articles/10.1051/cocv/2012033/} }

TY - JOUR AU - Caboussat, Alexandre AU - Glowinski, Roland AU - Sorensen, Danny C. TI - A least-squares method for the numerical solution of the Dirichlet problem for the elliptic monge-ampère equation in dimension two JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2013 SP - 780 EP - 810 VL - 19 IS - 3 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2012033/ DO - 10.1051/cocv/2012033 LA - en ID - COCV_2013__19_3_780_0 ER -

%0 Journal Article %A Caboussat, Alexandre %A Glowinski, Roland %A Sorensen, Danny C. %T A least-squares method for the numerical solution of the Dirichlet problem for the elliptic monge-ampère equation in dimension two %J ESAIM: Control, Optimisation and Calculus of Variations %D 2013 %P 780-810 %V 19 %N 3 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2012033/ %R 10.1051/cocv/2012033 %G en %F COCV_2013__19_3_780_0

Caboussat, Alexandre; Glowinski, Roland; Sorensen, Danny C. A least-squares method for the numerical solution of the Dirichlet problem for the elliptic monge-ampère equation in dimension two. ESAIM: Control, Optimisation and Calculus of Variations, Volume 19 (2013) no. 3, pp. 780-810. doi : 10.1051/cocv/2012033. http://www.numdam.org/articles/10.1051/cocv/2012033/

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