Finite element methods for fully nonlinear second order PDEs based on a discrete Hessian with applications to the Monge–Ampère equation

摘要

The purpose of this paper is twofold. First, we modify a method due to Lakkis and Pryer where the notion of a discrete Hessian is introduced to compute fully nonlinear second order PDEs. The discrete Hessian used in our approach is entirely local, making the resulting linear system within the Newton iteration much easier to solve. The second contribution of this paper is to analyze both Lakkis and Pryer’s method and its modification in parallel applied to the two-dimensional Monge–Ampère equation. In both cases we show the well-posedness of the methods as well as derive optimal error estimates. Numerical experiments are presented which (i) back up the theoretical findings and (ii) indicate that the methods are able to capture weak (viscosity) solutions.