Finite element approximations of general fully nonlinear second order elliptic partial differential equations based on the vanishing moment method

Abstract

The vanishing moment method was introduced by the authors in Feng and Neilan (2009) as a reliable methodology for computing viscosity solutions of fully nonlinear second order partial differential equations (PDEs). It is based on the simple idea of approximating a "hard-to-handle" fully nonlinear second order PDE by a family (parameterized by a small parameter ε) of "easy-to-handle" quasilinear fourth order PDEs. The primary objective of this article is to present a comprehensive finite element analysis for the vanishing moment approximation of general fully nonlinear second order elliptic PDEs which fulfill some structure conditions. Abstract methodological and convergence analysis frameworks of conforming finite element methods are first developed for fully nonlinear second order PDEs in a general setting. The abstract framework is then applied to three prototypical nonlinear equations, namely, the Monge-Ampère equation, the equation of prescribed Gauss curvature, and the infinity-Laplacian equation. Numerical experiments are presented for each problem to validate the theoretical error estimate results and to gauge the efficiency of the proposed numerical methods and the vanishing moment methodology.