基于消失矩方法的全非线性Monge-Ampère方程的混合有限元方法
Mixed finite element methods for the fully nonlinear Monge-Ampère equation based on the vanishing moment method
AI总结：
本文基于消失矩方法，通过四阶正则化方程逼近二阶全非线性Monge-Ampère方程，提出Hermann-Miyoshi型混合有限元方法，同时逼近解和矩张量，并给出误差估计和数值收敛性研究。
摘要：
本文研究了基于消失矩方法的全非线性Monge-Ampère方程$\det(D^2u^0)=f$的Dirichlet问题粘性解的混合有限元逼近，该方法由作者在\cite{Feng2}中最近提出。在该方法中，二阶全非线性Monge-Ampère方程由四阶拟线性方程$-εΔ^2 u^ε+ \det{D^2u^ε} =f$逼近。文献\cite{Feng1}证明了$u^ε$收敛到Monge-Ampère方程Dirichlet问题的唯一凸粘性解$u^0$。这一结果为构造全非线性二阶方程的收敛有限元方法打开了大门，而此前这一任务难以实现。本文的目标有三：首先，我们发展了一族Hermann-Miyoshi型混合有限元方法来逼近正则化四阶问题的解$u^ε$，同时计算$u^\vepsi$和矩张量$σ^\vepsi:=D^2u^ε$。其次，我们推导了误差估计，显式追踪误差常数对参数$\vepsi$的依赖性，针对误差$u^ε-u^ε_h$和$σ^\vepsi-σ_h^\vepsi$。最后，我们给出了关于误差$u^0-u_h^\vepsi$和$σ^\vepsi-σ_h^\vepsi$的$\vepsi$幂次收敛速率的详细数值研究，并数值检验了为实现这些速率，网格尺寸$h$与$\vepsi$之间的“最佳”关系。
英文摘要：
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.
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