Metric based up-scaling

Heterogeneous multi-scale structures can be found everywhere in nature. Can these structures be accurately simulated at a coarse level? Homogenization theory allows us to do so under the assumptions of ergodicity and scale separation by transferring bulk (averaged) information from sub-grid scales to computational scales. Can we get rid of these assumptions? Can we compress a PDE with arbitrary coefficients? Surprisingly the answer is yes; it is rigorous and based on a new form of compensation. We will consider divergence form elliptic operators in dimension $n\geq 2$ to introduce this method. Although solutions of these operators are only H\"{o}lder continuous, we show that their regularity with respect to Harmonic mappings is $C^{1,\alpha}$. It follows that these PDEs can be up-scaled by transferring a new metric in addition to traditional bulk quantities from small scales into coarse scales and error bounds can be given.