The locally adapted patch finite element method for interface problems   on triangular meshes
  We present a locally adapted parametric finite element method for interface problems. For this adapted finite element method we show optimal convergence for elliptic interface problems with a discontinuous diffusion parameter. The method is based on the adaption of macro elements where a local basis represents the interface. The macro elements are independent of the interface and can be cut by the interface. A macro element which is a triangle in the triangulation is divided into four subtriangles. On these subtriangles, the basis functions of the macro element are interpreted as linear functions. The position of the vertices of these subtriangles is determined by the location of the interface in the case a macro element is cut by the interface. Quadrature is performed on the subtriangles via transformations to a reference element. Due to the locality of the method, its use is well suited on distributed architectures. 
Johan Hoffman, B\"arbel Holm and Thomas Richter
math.NA