[Paper Review] A finite element method for the Monge-Ampère equation with transport boundary conditions

Ellya L. Kawecki, Omar Lakkis+1 more|arXiv (Cornell University)|Jul 10, 2018

Geometric Analysis and Curvature Flows16 references8 citations

TL;DR

This paper presents a nonvariational finite element method (NVFEM) for solving the Monge-Ampère equation with optimal transport boundary conditions, using Newton–Raphson linearization and gradient/Hessian recovery for optimal convergence. The method achieves empirically optimal convergence rates in L2 and H1 norms for piecewise polynomial approximations, validated through numerical experiments in optics and mesh movement.

ABSTRACT

We address the numerical solution via Galerkin type methods of the Monge-Ampère equation with transport boundary conditions arising in optimal mass transport, geometric optics and computational mesh or grid movement techniques. This fully nonlinear elliptic problem admits a linearisation via a Newton-Raphson iteration, which leads to an oblique derivative boundary value problem for elliptic equations in nondivergence form. We discretise these by employing the nonvariational finite element method, which lead to empirically observed optimal convergence rates, provided recovery techinques are used to approximate the gradient and the Hessian of the unknown functions. We provide extensive numerical testing to illustrate the strengths of our approach and the potential applications in optics and mesh movement.

Motivation & Objective

• To develop a Galerkin-type finite element method for the fully nonlinear Monge-Ampère equation with optimal transport boundary conditions.
• To address the challenge of solving nondivergence form elliptic equations arising from Newton–Raphson linearization of the Monge-Ampère PDE.
• To enable robust and accurate numerical solution using nonvariational finite elements with gradient and Hessian recovery techniques.
• To demonstrate applicability in geometric optics and adaptive mesh movement through numerical experiments.
• To provide a computationally efficient, easily implementable method using standard finite element packages like FEniCS.

Proposed method

• The method employs Newton–Raphson iteration to linearize the fully nonlinear Monge-Ampère equation into a sequence of nondivergence form elliptic problems with oblique derivative boundary conditions.
• A nonvariational finite element method (NVFEM) is used to discretize the linearized problems, avoiding weak formulations and enabling conforming approximations on unstructured meshes.
• Gradient and Hessian recovery techniques are applied to reconstruct the gradient and Hessian of the solution, which are essential for approximating the transport map ∇u.
• The method uses piecewise polynomial finite element spaces (P1, P2, P3) for the solution and its derivatives, with recovery enhancing convergence rates.
• Boundary conditions are enforced via the transport condition ∇u(∂Ω) = ∂Υ, equivalent to t(∂Ω) = ∂Υ under convexity assumptions.
• The approach is implemented using the FEniCS finite element package, enabling straightforward adaptation and high-order extensions.

Experimental results

Research questions

• RQ1 Can a Galerkin-type finite element method achieve optimal convergence for the Monge-Ampère equation with optimal transport boundary conditions?
• RQ2 How effective is gradient and Hessian recovery in restoring optimal convergence rates in nonvariational finite element schemes?
• RQ3 Can the NVFEM be successfully applied to nonlinear elliptic problems with oblique derivative boundary conditions arising from optimal transport?
• RQ4 What is the performance of the method in practical applications such as geometric optics and mesh movement?
• RQ5 Is the method robust and easily implementable using standard finite element software?

Key findings

• The method achieves optimal convergence rates in the L2 and H1 norms for P1, P2, and P3 finite elements when gradient recovery is applied, with observed convergence orders of approximately 1.8–1.9 for H1 and L2 errors.
• Without gradient recovery, convergence is suboptimal or fails to converge for P1 elements, highlighting the necessity of recovery techniques.
• For P2 and P3 elements, the observed convergence orders for the Hessian and gradient approximations are close to optimal, with eoc values of 1.85 and 1.93 respectively.
• The method successfully generates monitor-function-based grids for image processing, as demonstrated by the Gaspard Monge portrait example, where mass transport distorts a uniform mesh to match a target density.
• The approach enables accurate approximation of the transport map ∇u via recovered gradients, which is critical for applications in optics and mesh adaptation.
• The computational framework is publicly available, facilitating reproducibility and extension to higher-order isoparametric elements.