Analysis of Galerkin Methods for the Fully Nonlinear Monge-Ampère Equation

This paper develops and analyzes finite element Galerkin and spectral Galerkin methods for approximating viscosity solutions of the fully nonlinear Monge-Ampère equation det(D2u0) = f (> 0) based on the vanishing moment method which was developed by the authors in [17, 15]. In this approach, the Monge-Ampère equation is approximated by the fourth order quasilinear equation −ε∆2uε + det D2uε = f accompanied by appropriate boundary conditions. This new approach allows one to construct convergent Galerkin numerical methods for the fully nonlinear Monge-Ampère equation (and other fully nonlinear second order partial differential equations), a task which has been impracticable before. In this paper, we first develop some finite element and spectral Galerkin methods for approximating the solution uε of the regularized fourth order problem. We then derive optimal order error estimates for the proposed numerical methods. In particular, we track explicitly the dependence of the error bounds on the parameter ε, for the error uε − uh. Due to the strong nonlinearity of the underlying equation, the standard perturbation argument for error analysis of finite element approximations of nonlinear problems does not work here. To overcome the difficulty, we employ a fixed point technique which strongly makes use of the stability property of the linearized problem and its finite element approximations. Finally, using the Aygris finite element method as an example, we present a detailed numerical study of the rates of convergence in terms of powers of ε for the error u0 − uh, and numerically examine what is the “best” mesh size h in relation to ε in order to achieve these rates.