Pointwise rates of convergence for the Oliker-Prussner method for the Monge-Ampère equation

Abstract

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.

33 pages, 3 figures

Cited by in corpus (7)

• Quantitative stability of optimal transport maps and linearization of the 2-Wasserstein space
• Monotone meshfree methods for linear elliptic equations in non-divergence form via nonlocal relaxation
• Two-scale method for the Monge-Ampère Equation: Pointwise Error Estimates
• Two-Scale Method for the Monge-Ampère Equation: Convergence to the Viscosity Solution
• Convergence of a damped Newton's method for discrete Monge-Ampere functions with a prescribed asymptotic cone
• Rates of convergence in $W_p^2$-norm for the Monge-Ampère equation
• Numerical analysis of strongly nonlinear PDEs