Volume 22, pp. 71-96, 2006.

An augmented Lagrangian approach to the numerical solution of the Dirichlet problem for the elliptic Monge-Ampère equation in two dimensions

E. J. Dean and R. Glowinski

Abstract

In this article, we discuss the numerical solution of the Dirichlet problem for the real elliptic Monge-Ampère equation, in two dimensions, by an augmented Lagrangian based iterative method. To derive the above algorithm, we take advantage of a reformulation of the Monge-Ampère problem as a saddle-point one, for a well-chosen augmented Lagrangian functional over the product of appropriate primal and dual sets. The convergence of the finite element approximation and of the iterative methods described in this article still has to be proved, however, on the basis of numerical experiments reported in this article, it is safe to say that: (i) The augmented Lagrangian methodology discussed here provides a sequence converging to a solution of the Monge-Ampère problem under consideration, if such a solution exists in the space $H^2(\mathrm{\Omega })$. (ii) If, despite the smoothness of its data, the above problem has no solution, the above augmented Lagrangian method solves it in a least-squares sense, to be precisely defined in this article.

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Key words

elliptic Monge-Ampère equation, augmented Lagrangian algorithms, mixed finite element approximations

AMS subject classifications

35J60, 65F10, 65N30