Fully discrete least-squares splitting scheme for the Monge-Ampère equation: finite element analysis and convergence

The least-squares splitting algorithm for the Monge-Amp\`ere equation has been used successfully in computations for several years, but a convergence theory for fully discrete splitting schemes of this type has remained unavailable. In this work, we introduce and analyze a finite element framework for smooth solutions of the Dirichlet Monge-Amp\`ere equation in two dimensions. The proposed schemes combine a discrete Hessian reconstruction with a local projection onto the determinant constraint. Under a discrete Miranda-Talenti estimate and standard approximation properties of the Hessian reconstruction, we prove local convergence of the iterative scheme and optimal-order convergence of its limit to the exact solution in an $H^2$-type norm. We verify the estimates for conforming $C^1$ schemes, including the Argyris element, and for $C^0$-interior penalty and DG schemes of degree at least three; quadratic $C^0$-interior penalty and DG schemes are also covered when $|u|_{H^3(\Omega)}$ is sufficiently small. To the best of our knowledge, these discretizations have not previously been proposed or analyzed for least-squares splitting methods. Numerical experiments confirm the theoretical convergence rates.