Communications in Mathematical Sciences

Volume 16 (2018)

Number 7

Approximate homogenization of convex nonlinear elliptic PDEs

Pages: 1985 – 1906

DOI: http://dx.doi.org/10.4310/CMS.2018.v16.n7.a7

Authors

Chris Finlay (Department of Mathematics and Statistics, McGill University, Montreal, Quebec, Canada)

Adam M. Oberman (Department of Mathematics and Statistics, McGill University, Montreal, Quebec, Canada)

Abstract

We approximate the homogenization of fully nonlinear, convex, uniformly elliptic partial differential equations in the periodic setting, using a variational formula for the optimal invariant measure, which may be derived via Legendre–Fenchel duality. The variational formula expresses $\overline{H}(Q)$ as an average of the operator against the optimal invariant measure, generalizing the linear case. Several nontrivial analytic formulas for $\overline{H}(Q)$ are obtained. These formulas are compared to numerical simulations, using both PDE and variational methods. We also perform a numerical study of convergence rates for homogenization in the periodic and random setting and compare these to theoretical results.

Keywords

elliptic partial differential equations, homogenization, finite difference schemes, Pucci operator

2010 Mathematics Subject Classification

35J70, 52A41, 65N06, 93E20

Full Text (PDF format)

Received 3 November 2017

Accepted 28 May 2018