Contents Online
Communications in Mathematical Sciences
Volume 16 (2018)
Number 7
Approximate homogenization of convex nonlinear elliptic PDEs
Pages: 1985 – 1906
DOI: https://dx.doi.org/10.4310/CMS.2018.v16.n7.a7
Authors
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
Received 3 November 2017
Accepted 28 May 2018
Published 7 March 2019