BCAM Seminar Homogenization approximations for PDEs with non-separated scales and high contrast

Numerical homogenization for problems with multiple scales have attracted increasing attention in recent years. In particular, problems with non-separable scales pose a great challenge to mathematical analysis and simulation. Most existing methods are either based on the assumption of scale separation or heuristic arguments.

In this talk, we present some rigorous results on homogenization of scalar and vectorial partial differential equations with $L^\\infty$ coefficients which allow for a continuum of spatial and temporal scales. The first approach is based on a new type of compensation phenomena for scalar partial differential equations using the so-called \harmonic coordinates\". The second approach, namely, the flux norm approach can be applied to finite dimensional homogenization approximations of both vectorial and scalar problems with non-separated scales and high contrast. Numerical methods are developed and analyzed for both approaches. We will also discuss the connection and application of our methods to atomistic to continuum problems."