Mixed Finite Element Methods for the Fully Nonlinear Monge-Ampère Equation Based on the Vanishing Moment Method

• Authors: Xiaobing Feng;Michael Neilan
• Venue: SIAM Journal on Numerical Analysis
• Year: 2009

This paper studies mixed finite element approximations of the viscosity solution to the Dirichlet problem for the fully nonlinear Monge-Ampère equation $\det(D^2u^0)=f\,(0)$ based on the vanishing moment method which was proposed recently by the authors in [X. Feng and M. Neilan, J. Scient. Comp., DOI 10.1007/s10915-008-9221-9, 2008]. In this approach, the second-order fully nonlinear Monge-Ampère equation is approximated by the fourth order quasilinear equation $-\varepsilon\Delta^2 u^\varepsilon + \det{D^2u^\varepsilon}=f$. It was proved in [X. Feng, Trans. AMS, submitted] that the solution $u^\varepsilon$ converges to the unique convex viscosity solution $u^0$ of the Dirichlet problem for the Monge-Ampère equation. This result then opens a door for constructing convergent finite element methods for the fully nonlinear second-order equations, a task which has been impracticable before. The goal of this paper is threefold. First, we develop a family of Hermann-Miyoshi-type mixed finite element methods for approximating the solution $u^\varepsilon$ of the regularized fourth-order problem, which computes simultaneously $u^\varepsilon$ and the moment tensor $\sigma^\varepsilon:=D^2u^\varepsilon$. Second, we derive error estimates, which track explicitly the dependence of the error constants on the parameter $\varepsilon$, for the errors $u^\varepsilon-u^\varepsilon_h$ and $\sigma^0-\sigma_h^\varepsilon$. Finally, we present a detailed numerical study on the rates of convergence in terms of powers of $\varepsilon$ for the error $u^0-u_h^\varepsilon$ and $\sigma^\varepsilon-\sigma_h^\varepsilon$, and numerically examine what is the “best” mesh size $h$ in relation to $\varepsilon$ in order to achieve these rates. 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 for the problem. To overcome the difficulty, we employ a fixed point technique which strongly relies on the stability of the linearized problem and its mixed finite element approximations.