Adaptive composite finite elements for the solution of PDEs containing nonuniformely distributed micro-scales

In this paper we will introduce Adaptive Composite Finite Elements as a discrete homogenization technique for partial differential equations having small micro-structures as, e.g., rough boundaries or jumping coefficients. These Finite Elements allow to discretize such problems only with a few degrees of freedom and still getting the required asymptotic approximation property. This method can be applied for both, a relatively crude approximation of the PDE and the application of multi-grid methods to problems where standard finite elements would always result in systems of equations having a huge number of unknowns.