Nonconforming virtual element method for the Monge-Ampère equation
This work provides a novel numerical method for solving the fully nonlinear Monge-Ampère equation, which is important for applications in geometry and fluid dynamics.
The authors developed a C1-nonconforming C0-conforming virtual element method for the vanishing moment approximation of the Monge-Ampère equation, deriving optimal a priori error estimates and validating convergence rates numerically.
In this article, we develop the $C^1$-nonconforming $C^0$-conforming virtual element method (VEM) for the vanishing moment approximation of the second-order fully nonlinear Monge-Ampère equation in two dimensions. In the vanishing moment equation an artificial biharmonic term is introduced which produces a quasilinear fourth order problem. We derive optimal a priori error estimates in the $H^2$-, $H^1$- and $L^2$-norms for the virtual element method, and show the existence and uniqueness of the virtual element solution. We perform several numerical experiments to validate the convergence rate of the error with respect to the mesh size.