Mixed finite element methods for the fully nonlinear Monge-Amp\`ere equation based on the vanishing moment method

This paper studies mixed finite element approximations of the viscosity solution to the Dirichlet problem for the fully nonlinear Monge-Amp`ere equation $\det (D^2{u}^0)=f$ 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-Amp`ere equation is approximated by the fourth order quasilinear equation $-ϵ{\mathrm{\Delta }}^2{u}^{ϵ}+\det D^2{u}^{ϵ}=f$. 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-Amp`ere 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 $u^{\text{vepsi}}$ and the moment tensor ${\sigma }^{\text{vepsi}}:=D^2{u}^{ϵ}$. Second, we derive error estimates, which track explicitly the dependence of the error constants on the parameter $\text{vepsi}$, for the errors $u^{ϵ}-u_h^{ϵ}$ and ${\sigma }^{\text{vepsi}}-{\sigma }_h^{\text{vepsi}}$. Finally, we present a detailed numerical study on the rates of convergence in terms of powers of $\text{vepsi}$ for the error $u^0-u_h^{\text{vepsi}}$ and ${\sigma }^{\text{vepsi}}-{\sigma }_h^{\text{vepsi}}$, and numerically examine what is the “best” mesh size $h$ in relation to $\text{vepsi}$ in order to achieve these rates.