$\mathcal {C}^0$ penalty methods for the fully nonlinear Monge-Ampère equation

Abstract:

In this paper, we develop and analyze $\mathcal {C}^0$ penalty methods for the fully nonlinear Monge-Ampère equation $\det (D^2 u) = f$ in two dimensions. The key idea in designing our methods is to build discretizations such that the resulting discrete linearizations are symmetric, stable, and consistent with the continuous linearization. We are then able to show the well-posedness of the penalty method as well as quasi-optimal error estimates using the Banach fixed-point theorem as our main tool. Numerical experiments are presented which support the theoretical results.