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

Abstract

Abstract. This paper develops and analyzes finite element Galerkin and spectral Galerkin methods for approximating viscosity solutions of the fully nonlinear Monge-Ampère equation det(D 2 u 0 ) = 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∆ 2 u ε + det D 2 u ε = 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 u ε 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 u ε − u ε h . 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 u 0 − u ε h , and numerically examine what is the "best" mesh size h in relation to ε in order to achieve these rates.

Two Numerical Methods for the elliptic Monge-Ampère equation

“…Feng and Neilan [12][13][14] solve second order equations (including the Monge-Ampère equation) by adding a small multiple of the bilaplacian. The bilaplacian term introduces an additional discretization error and additional boundary conditions, which may not be compatible with the solution.…”

“…In particular, in the case of [7][8][9][10]16], the methods are known to fail. In the case of [12][13][14] we are not aware of computational results.…”

“…Feng and Neilan [12][13][14] do not give details about the computation time or iteration count of their numerical method.…”

“…The numerical solution of the elliptic Monge-Ampère PDE has been a subject of increasing interest [7][8][9][10][12][13][14]16,18,21,22], including an invited lecture at 2007 ICIAM [15].…”

Convergence Rate of a Stable, Monotone and Consistent Scheme for the Monge-Ampère Equation

“…Rate of convergence for smooth solutions were previously established in the context of finite elements [3][4][5] or the standard finite difference method [6,7]. The rate of convergence proven in this paper for a stable, monotone and consistent scheme is a key component of the theory developed in [7] for the convergence of finite difference discretizations to the Aleksandrov solution of the Monge-Ampère equation.…”