The numerical solution of the Dirichlet problem for an elliptic Pucci’s equation in two dimensions of space is addressed by using a least-squares approach. The algorithm relies on an iterative relaxation method that decouples a variational linear elliptic PDE problem from the local nonlinearities. The approximation method relies on mixed low order finite element methods. The least-squares framework allows to revisit and extend the approach and the results presented in [Caffarelli, Glowinski, 2008] to more general cases. Numerical results show the convergence of the iterative sequence to the exact solution, when such a solution exists. The robustness of the approach is highlighted, when dealing with various types of meshes, domains with curved boundaries, nonconvex domains, or non-smooth solutions.