Mixed finite element methods for the fully nonlinear Monge-Ampère equation based on the vanishing moment method

Abstract

This paper studies mixed finite element approximations of the viscosity solution to the Dirichlet problem for the fully nonlinear Monge-Ampère equation $\det(D^2u^0)=f$ based on the vanishing moment method which was proposed recently by the authors in \cite{Feng2}. In this approach, the second order fully nonlinear Monge-Ampère equation is approximated by the fourth order quasilinear equation $-εΔ^2 u^ε+ \det{D^2u^ε} =f$. It was proved in \cite{Feng1} that the solution $u^ε$ 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 $u^ε$ of the regularized fourth order problem, which computes simultaneously $u^\vepsi$ and the moment tensor $σ^\vepsi:=D^2u^ε$. Second, we derive error estimates, which track explicitly the dependence of the error constants on the parameter $\vepsi$, for the errors $u^ε-u^ε_h$ and $σ^\vepsi-σ_h^\vepsi$. Finally, we present a detailed numerical study on the rates of convergence in terms of powers of $\vepsi$ for the error $u^0-u_h^\vepsi$ and $σ^\vepsi-σ_h^\vepsi$, and numerically examine what is the "best" mesh size $h$ in relation to $\vepsi$ in order to achieve these rates.