Numerical Solution of the Simple Monge-Ampère Equation with Non-convex Dirichlet Data on Non-convex Domains
For researchers in numerical PDEs, this work shows empirically that convergence holds under weaker conditions than previously proven, though the results are numerical rather than theoretical.
The paper provides numerical evidence that the semi-Lagrangian method for the simple Monge-Ampère equation converges to viscosity solutions even on non-convex domains with non-convex Dirichlet data, extending previous theoretical results that required convexity.
The existence of a unique numerical solution of the semi-Lagrangian method for the simple Monge-Ampère equation is known independently of the convexity of the domain or Dirichlet boundary data -- when the Monge-Ampère equation is posed as Bellman problem. However, the convergence to the viscosity solution has only been proved on strictly convex domains. In this paper we provide numerical evidence that convergence of numerical solutions is observed more generally without convexity assumptions. We illustrate how in the limit multi-valued functions may be approximated to satisfy the Dirichlet conditions on the boundary as well as local convexity in the interior of the domain.