Numerical solution of the two-dimensional elliptic Monge–Ampère equation with Dirichlet boundary conditions: a least-squares approach

Abstract

We addressed, in a previous note [C. R. Acad. Sci. Paris, Ser. I 336 (2003) 779–784], the numerical solution of the Dirichlet problem for the two-dimensional elliptic Monge–Ampère equation, namely: det D 2 ψ = f in Ω, ψ = g on ∂ Ω ( Ω ⊂ R 2 and f > 0 , here). The method discussed previously relies on an augmented Lagrangian algorithm operating in the space H 2 ( Ω ) and related functional spaces of symmetric tensor-valued functions. In the particular case where the above problem has no solution in H 2 ( Ω ) , while the data f and g verify { f , g } ∈ L 1 ( Ω ) × H 3 / 2 ( ∂ Ω ) , there is strong evidence that the augmented Lagrangian algorithm discussed in previously converges-in some sense-to a least squares solution belonging to V g = { φ | φ ∈ H 2 ( Ω ) , φ = g on ∂ Ω } . Our goal in this note is to discuss a least-squares based alternative solution method for the Monge–Ampère Dirichlet problem. This method relies on the minimization on the set V g × Q f (with Q f = { q | q = ( q i j ) 1 ⩽ i , j ⩽ 2 , q i j ∈ L 2 ( Ω ) , ∀ i , j , 1 ⩽ i , j ⩽ 2 , q = q t , det q = f } ) of a well-chosen least-squares functional. From a practical point of view we solve the above minimization problem via a relaxation type algorithm, operating alternatively in V g and Q f and very easy to combine to the mixed finite element approximations employed in the earlier work. Numerical experiments show that the above method has good convergence properties when the Monge–Ampère Dirichlet problem has solutions in V g ; they show also that, for cases where the above problem has no solution in V g , while neither V g nor Q f are empty, the new method reproduces the solutions obtained via the augmented Lagrangian approach, but faster. To cite this article: E.J. Dean, R. Glowinski, C. R. Acad. Sci. Paris, Ser. I 339 (2004).