Multiscale algorithm with patches of finite elements

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. We discuss the implementation and illustrate the method on an industrial example.