The composite mini element: a new mixed FEM for the Stokes equations on complicated domains

Abstract

We introduce a new finite element method, the composite mini element , for the mixed discretization of the Stokes equations on two and three‐dimensional domains that may contain a huge number of geometric details. Instead of a geometric resolution of the domain and the boundary condition by the finite element mesh the shape of the finite element functions is adapted to the geometric details. This approach allows low‐dimensional approximations even for problems with complicated geometric details such as holes or rough boundaries. It turns out that the method can be viewed as a coarse scale generalization of the classical mini element approach, i.e. it reduces the computational effort while the approximation quality depends linearly on the (coarse) mesh size in the usual way. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)