Approximate homogenization of fully nonlinear elliptic PDEs: estimates and numerical results for Pucci type equations

TL;DR

This paper develops approximate homogenization techniques for fully nonlinear elliptic PDEs of Pucci type, providing error estimates and numerical validation showing high accuracy of linearization in various regimes.

Contribution

It introduces a new error estimation method for linearization of nonlinear Pucci-type PDEs and demonstrates its effectiveness through numerical experiments.

Findings

Linearization yields accurate approximations away from high curvature regions.

Numerical results show errors are often just a few percent.

Even near corners, the linearization remains highly accurate.

Abstract

We are interested in the shape of the homogenized operator $\overline F(Q)$ for PDEs which have the structure of a nonlinear Pucci operator. A typical operator is $H^{a_1,a_2}(Q,x) = a_1(x) \lambda_{\min}(Q) + a_2(x)\lambda_{\max}(Q)$. Linearization of the operator leads to a non-divergence form homogenization problem, which can be solved by averaging against the invariant measure. We estimate the error obtained by linearization based on semi-concavity estimates on the nonlinear operator. These estimates show that away from high curvature regions, the linearization can be accurate. Numerical results show that for many values of $Q$, the linearization is highly accurate, and that even near corners, the error can be small (a few percent) even for relatively wide ranges of the coefficients.