Convergent semi-Lagrangian methods for the Monge-Amp\`ere equation on unstructured grids

This paper is concerned with developing and analyzing convergent semi-Lagrangian methods for the fully nonlinear elliptic Monge-Amp\`ere equation on general triangular grids. This is done by establishing an equivalent (in the viscosity sense) Hamilton-Jacobi-Bellman formulation of the Monge-Amp\`ere equation. A significant benefit of the reformulation is the removal of the convexity constraint from the admissible space as convexity becomes a built-in property of the new formulation. Moreover, this new approach allows one to tap the wealthy numerical methods, such as semi-Lagrangian schemes, for Hamilton-Jacobi-Bellman equations to solve Monge-Amp\`ere type equations. It is proved that the considered numerical methods are monotone, pointwise consistent and uniformly stable. Consequently, its solutions converge uniformly to the unique convex viscosity solution of the Monge-Amp\`ere Dirichlet problem. A super-linearly convergent Howard's algorithm, which is a Newton type method, is utilized as the nonlinear solver to take advantage of the monotonicity of the scheme. Numerical experiments are also presented to gauge the performance of the proposed numerical method and the nonlinear solver.