In this paper we construct a numerical homogenization technique for nonlinear elliptic equations. In particular, we are interested in when the elliptic flux depends on the gradient of the solution in a nonlinear fashion which makes the numerical homogenization procedure nontrivial. The convergence of the numerical procedure is presented for the general case using G-convergence theory. To calculate the fine scale oscillations of the solutions we propose a stochastic two-scale corrector where one of the scales is a numerical scale and the other is a physical scale. The analysis of the convergence of two-scale correctors is performed under the assumption that the elliptic flux is strictly stationary with respect to spatial variables. The nonlinear multiscale finite element method has been proposed and analyzed.
|Original language||English (US)|
|Number of pages||18|
|Journal||Multiscale Modeling and Simulation|
|State||Published - 2004|
Bibliographical notePublisher Copyright:
© 2003 Society for Industrial and Applied Mathematics.
- Finite element
ASJC Scopus subject areas
- Modeling and Simulation
- Ecological Modeling
- Physics and Astronomy(all)
- Computer Science Applications