A least-squares method for the numerical solution of the Dirichlet problem for the elliptic monge-ampère equation in dimension two

Caboussat, Alexandre; Glowinski, Roland; Sorensen, Danny C.

ESAIM: Control, Optimisation and Calculus of Variations, Tome 19 (2013) no. 3, pp. 780-810.

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.

Classification : 65N30, 65K10, 65F30, 49M15, 49K20
Mots clés : Monge − ampère equation, least-squares method, biharmonic problem, conjugate gradient method, quadratic constraint minimization, mixed finite element methods

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