  In this work, we describe a simple finite element approach that is able to
resolve weak discontinuities in interface problems accurately. The approach is
based on a fixed patch mesh consisting of quadrilaterals, that will stay
unchanged independent of the position of the interface. Inside the patches we
refine once more, either in eight triangles or in four quadrilaterals, in such
a way that the interface is locally resolved. The resulting finite element
approach can be considered a fitted finite element approach. In our practical
implementation, we do not construct this fitted mesh, however. Instead, the
local degrees of freedom are included in a parametric way in the finite element
space, or to be more precise in the local mappings between a reference patch
and the physical patches. We describe the implementation in the open source C++
finite element library deal.II in detail and present two numerical examples to
illustrate the performance of the approach. Finally, detailed studies of the
behavior of iterative linear solvers complement this work.
