Pointwise rates of convergence for the Oliker-Prussner method for the Monge-Ampère equation
Provides rigorous convergence guarantees for a numerical method for the Monge-Ampère equation, a challenging fully nonlinear PDE, but the result is incremental as it builds on existing geometric techniques.
The paper establishes pointwise convergence rates for the Oliker-Prussner method applied to the Monge-Ampère equation, proving exactness for convex quadratic polynomials under translation-invariant nodes and deriving stability via discrete Alexandroff and Brunn-Minkowski inequalities.
We study the Oliker-Prussner method exploiting its geometric nature. We derive discrete stability and continuous dependence estimates in the max-norm by using a discrete Alexandroff estimate and the Brunn-Minkowski inequality. We show that the method is exact for all convex quadratic polynomials provided the underlying set of nodes is translation invariant within the domain; nodes still conform to the domain boundary. This gives a suitable notion of operator consistency which, combined with stability, leads to pointwise rates of convergence for classical and non-classical solutions of the Monge-Ampère equation.