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

TL;DR

This paper develops mixed finite element methods based on the vanishing moment approach to approximate solutions of the fully nonlinear Monge-Ampère equation, providing error estimates and numerical convergence analysis.

Contribution

It introduces a new family of mixed finite element methods for the Monge-Ampère equation using the vanishing moment method, with explicit error estimates and numerical validation.

Findings

Error estimates depend explicitly on the regularization parameter psi

Numerical results confirm theoretical convergence rates

Optimal mesh size relative to psi is identified

Abstract

This paper studies mixed finite element approximations of the viscosity solution to the Dirichlet problem for the fully nonlinear Monge-Amp\`ere equation $\det(D^2u^0)=f$ 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-Amp\`ere equation is approximated by the fourth order quasilinear equation $-\epsilon\Delta^2 u^\epsilon + \det{D^2u^\epsilon} =f$. It was proved in \cite{Feng1} that the solution $u^\epsilon$ 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 $u^\epsilon$ of the regularized fourth order problem, which computes simultaneously $u^\vepsi$ and the moment tensor $\sigma^\vepsi:=D^2u^\epsilon$. Second, we derive error estimates, which track explicitly the dependence of the error constants on the parameter $\vepsi$, for the errors $u^\epsilon-u^\epsilon_h$ and $\sigma^\vepsi-\sigma_h^\vepsi$. 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 $\sigma^\vepsi-\sigma_h^\vepsi$, and numerically examine what is the "best" mesh size $h$ in relation to $\vepsi$ in order to achieve these rates.