Numerical methods for fully nonlinear elliptic equations of the Monge–Ampère type

In this article, we discuss the numerical solution of the Dirichlet problem for the Monge–Ampère equation in two dimensions. The solution of closely related problems is also discussed; these include a family of Pucci’s equations, the equation prescribing the harmonic mean of the eigenvalues of the Hessian of a smooth function of two variables, and a minimization problem from nonlinear elasticity, where the cost functional involves the determinant of the gradient of vector-valued functions. To solve the Monge–Ampère equation we consider two methods. The first one “reduces”the Monge–Ampère equation to a saddle-point problem for a well-chosen augmented Lagrangian; to solve this saddle-point problem we advocate an Uzawa–Douglas–Rachford algorithm. The second method combines nonlinear least-squares and operator-splitting. This second method being simpler to implements, we apply variants of it to the solution of the other problems. For the space discretization we use mixed finite element approximations, closely related to methods already used for the solution of linear and nonlinear bi-harmonic problems; through these approximations the solution of the above problems is, essentially, reduced to the solution of discrete Poisson problems. The methods discussed in this article are validated by the results of numerical experiments.