Rates of convergence in $W^2_p$-norm for the Monge-Ampère equation
This provides rigorous convergence rates for a numerical method for the Monge-Ampère equation, a challenging fully nonlinear PDE.
The authors developed discrete W^2_p-norm error estimates for the Oliker-Prussner method for the Monge-Ampère equation, achieving rates of C h^{1/p} for p > d and C h^{1/d} (ln(1/h))^{1/d} for p ≤ d, with numerical examples confirming sharpness in 2D.
We develop discrete $W^2_p$-norm error estimates for the Oliker-Prussner method applied to the Monge-Ampère equation. This is obtained by extending discrete Alexandroff estimates and showing that the contact set of a nodal function contains information on its second order difference. In addition, we show that the size of the complement of the contact set is controlled by the consistency of the method. Combining both observations, we show that the error estimate $\|u - u_h\|_{W^2_p} \leq C h^{1/p}$ if $p > d$ and $\|u - u_h\|_{W^2_p} \leq C h^{1/d} \big(\ln\left(\frac 1 h \right)\big)^{1/d} $ if $p \leq d$. Here the constant $C$ depends on $\|{u}\|_{C^{3,1}(\barΩ)}$, the dimension $d$, and the constant $p$. Numerical examples are given in two space dimensions and confirm that the estimate is sharp in several cases.