Section: Application Domains

Numerical simulation in heterogeneous media

Solving numerically PDEs in highly heterogeneous media is a problem encountered in many situations, such as the transport of pollutants or the design of oil extraction strategies in geological undergrounds. When such problems are discretized by standard numerical methods the number of degrees of freedom may become prohibitive in practice, whence the need for other strategies.

Numerical solution methods inspired by asymptotic analysis are among the very few feasible alternatives, and started fifteen years ago with the contributions of Hou and Wu [59] , Arbogast [40] etc. We refer to [54] , [70] ,[3] for a recent state of the art. Numerical homogenization methods usually amount to looking for the solution of the problem (1 ) in the form $u_{\epsilon }\left(x\right)\simeq u_0\left(x\right)+\epsilon \nabla u_0\left(x\right)·\Phi (x,\frac{x}{\epsilon })$, where $\Phi$ is a proxy for the corrector field computed locally at point $x$ (in particular, one does not use explicitly that the problem is periodic so that the method can be used for more general coefficients) and $u_0$ is a function which does not oscillate at scale.

Relying on our quantitative insight in stochastic homogenization, a first task consists in addressing the three following prototypical academic examples: periodic, quasi-periodic, and stationary ergodic coefficients with short range dependence. The more ambitious challenge is to address more complex coefficients (of interest to practioners), and design adaptive and efficient algorithms for diffusion in heterogeneous media.