A finite element method for the Monge-Ampère equation with transport boundary conditions
This provides a numerical solution for a fully nonlinear elliptic PDE arising in optimal transport, optics, and mesh generation, but the results are empirical without theoretical convergence guarantees.
The paper develops a finite element method for the Monge-Ampère equation with transport boundary conditions, achieving empirically observed optimal convergence rates through Newton-Raphson linearization and nonvariational discretization with recovery techniques.
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.