We develop a discretization and solution technique for elliptic problems whose solutions may present strong variations, singularities, boundary layers and oscillations in localized regions. We start with a coarse finite element discretization with a mesh size H, and we superpose to it local patches of finite elements with finer mesh size h << H to capture local behaviour of the solution. The two meshes (coarse and patch) are not necessarily compatible. Similar to mesh adaptation methods, the location of the fine patches is identified by an a posteriori error estimator. Unlike mesh adaptation, no remeshing is involved. We discuss the implementation and illustrate the method on an industrial example. Copyright (C) 2007 John Wiley & Sons, Ltd.