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The existence of a unique numerical solution of the semi-Lagrangian methodfor the simple Monge-Amp\`ere equation is known independently of the convexityof the domain or Dirichlet boundary data -- when the Monge-Amp\`ere equation isposed as Bellman problem. However, the convergence to the viscosity solutionhas only been proved on strictly convex domains. In this paper we providenumerical evidence that convergence of numerical solutions is observed moregenerally without convexity assumptions. We illustrate how in the limitmulti-valued functions may be approximated to satisfy the Dirichlet conditionson the boundary as well as local convexity in the interior of the domain.

Numerical Solution of the Simple Monge-Amp\`ere Equation with Non-convex Dirichlet Data on Non-convex Domains