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

On the implementation of a locally modified finite element method for interface problems in deal.II