[Paper Review] Galerkin Methods for the Fully Nonlinear Monge-Ampère Equation

Xiaobing Feng, Michael Neilan|ArXiv.org|Dec 7, 2007

Nonlinear Partial Differential Equations14 references3 citations

TL;DR

This paper develops and analyzes Galerkin finite element and spectral methods for the fully nonlinear Monge-Ampère equation using the vanishing moment method, which regularizes the equation into a fourth-order quasilinear problem. The key contribution is optimal error estimates with explicit dependence on the regularization parameter $\varepsilon$, and numerical evidence shows that $h = \varepsilon^{1/2}$ yields optimal convergence rates for the global error $u^0 - u_h^\varepsilon$. The method overcomes the lack of standard perturbation techniques via a fixed-point argument leveraging linearized problem stability.

ABSTRACT

This paper develops and analyzes finite element Galerkin and spectral Galerkin methods for approximating viscosity solutions of the fully nonlinear Monge-Ampère equation $\det(D^2u^0)=f$ based on the vanishing moment method which was developed by the authors in \cite{Feng2,Feng1}. In this approach, the Monge-Ampère equation is approximated by the fourth order quasilinear equation $-εΔ^2 u^ε+ \det{D^2u^ε} =f$ accompanied by appropriate boundary conditions. This new approach allows one to construct convergent Galerkin numerical methods for the fully nonlinear Monge-Ampère equation, a task which has been impracticable before. In this paper, we first develop some finite element and spectral Galerkin methods for approximating the solution $u^ε$ of the regularized fourth order problem. We then derive optimal order error estimates for the proposed numerical methods. In particular, we track explicitly the dependence of the error bounds on the parameter $\vepsi$, for the error $u^ε-u^ε_h$. Finally, using the Aygris finite element method as an example, we present a detailed numerical study of the rates of convergence in terms of powers of $\vepsi$ for the error $u^0-u_h^\vepsi$, and numerically examine what is the "best" mesh size $h$ in relation to $\vepsi$ in order to achieve these rates.

Motivation & Objective

• To develop convergent Galerkin numerical methods for the fully nonlinear Monge-Ampere equation, which had been impracticable before due to strong nonlinearity.
• To analyze finite element and spectral Galerkin methods for the regularized fourth-order problem $-\varepsilon\Delta^2 u^\varepsilon + \det(D^2 u^\varepsilon) = f$.
• To derive optimal error estimates for $u^\varepsilon - u_h^\varepsilon$ with explicit dependence on the regularization parameter $\varepsilon$.
• To identify the optimal relationship between mesh size $h$ and regularization parameter $\varepsilon$ that achieves the best convergence rate for the global error $u^0 - u_h^\varepsilon$.

Proposed method

• Applies the vanishing moment method to regularize the fully nonlinear Monge-Ampere equation into a fourth-order quasilinear PDE: $-\varepsilon\Delta^2 u^\varepsilon + \det(D^2 u^\varepsilon) = f$.
• Uses Galerkin finite element and spectral Galerkin methods to approximate the solution $u^\varepsilon$ of the regularized problem.
• Employs a fixed-point technique to overcome the failure of standard perturbation arguments, relying on stability of the linearized problem and its finite element approximation.
• Analyzes error in $L^2$, $H^1$, and $H^2$ norms, tracking explicit $\varepsilon$-dependence in error bounds.
• Uses the Aygris finite element method as a numerical example to study convergence rates in $h$ and $\varepsilon$.
• Performs numerical experiments with $u^0 = x^4 + y^2$ and $u^0 = 20x^6 + y^6$ to test convergence under different $h$-$\varepsilon$ relations.

Experimental results

Research questions

• RQ1 What is the optimal convergence rate of the Galerkin method for the regularized Monge-Ampere problem in terms of $h$ and $\varepsilon$?
• RQ2 How does the error $u^\varepsilon - u_h^\varepsilon$ depend on the regularization parameter $\varepsilon$?
• RQ3 Can a fixed-point argument be used to establish optimal error estimates when standard perturbation techniques fail due to strong nonlinearity?
• RQ4 What is the best choice of mesh size $h$ relative to $\varepsilon$ to achieve optimal convergence for the global error $u^0 - u_h^\varepsilon$?
• RQ5 Does the convergence rate of $u^0 - u_h^\varepsilon$ match that of $u^0 - u^\varepsilon$ when $h = \varepsilon^{1/2}$?

Key findings

• Optimal error estimates are derived for $u^\varepsilon - u_h^\varepsilon$ in $L^2$, $H^1$, and $H^2$ norms, with explicit dependence on $\varepsilon$.
• Numerical results show that the error converges faster than predicted by theory, with $\|u^\varepsilon - u_h^\varepsilon\|_{L^2} \approx O(h^6)$, $\|u^\varepsilon - u_h^\varepsilon\|_{H^1} \approx O(h^5)$, and $\|u^\varepsilon - u_h^\varepsilon\|_{H^2} \approx O(h^4)$ for $\varepsilon = 0.001$.
• The global error $\|u^0 - u_h^\varepsilon\|_{H^2}$ converges as $O(\varepsilon^{1/4})$ when $h = \varepsilon^{1/2}$, matching the rate of $\|u^0 - u^\varepsilon\|_{H^2}$.
• The global error $\|u^0 - u_h^\varepsilon\|_{L^2}$ converges as $O(\varepsilon)$ when $h = \varepsilon^{1/2}$, matching the rate of $\|u^0 - u^\varepsilon\|_{L^2}$.
• When $h = \varepsilon$, the $H^1$-error $\|u^0 - u_h^\varepsilon\|_{H^1}$ converges as $O(\varepsilon^{1/2})$, indicating suboptimal performance.
• The relation $h = \varepsilon^{1/2}$ is numerically confirmed as the optimal choice for achieving the best convergence rate of the global error across all norms.