A finite element method for the Monge-Amp\`ere equation with transport boundary conditions

We address the numerical solution via Galerkin type methods of the Monge-Amp\`ere 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.