In multimaterial problems, surface and interface effects have the potential to be of great importance. In the case of electromagnetics, this may be due to currents concentrating on surfaces, whereas in the case of heat transfer, it may be due to the quality of thermal contact between two materials. In cases where Lagrangian (moving) meshes are untenable due to large-scale deformation, we are often left with the alternative of using Eulerian (non-moving) meshes coupled with some sort of mixture model. For some types of physics, these mixture models are quite good, in others they leave much to be desired.
The conformal decomposition finite element method (CDFEM), which was primarily developed for stationary fluid interface problems, is one technique for handling sub-cell discontinuities. Strongly related to eXtended finite elements (XFEM) and adaptive mesh refinement (AMR), CDFEM allows for Lagrangian-like interfaces in the context of a (basically) Eulerian mesh. Unfortunately, since the physics "chooses" the location of the interface, the interface can lie precariously close to an element boundary on the underlying fixed mesh. This problem, known as a "sliver cut," can have a deleterious effect on the conditioning of the matrix and the performance of the linear solver.
Contact conditions across the interface can complicate this even further. As opposed to "perfect contact," where the solution is single-valued on the interface, problems can have "imperfect contact," where the solution has a different value on each side of the interface, but are coupled in some way. This can cause additional problems for multigrid coarsening, if these discontinuities aren't handled with care.
We present a local smoothing approach for CDFEM matrices, which uses information about the problem geometry near the interface to perform a single geometric coarsening to a problem with volume-averaged materials. As the coarse problem now lacks sliver cuts, it can be easily treated with standard algebraic multigrid techniques. We compare the performance of this technique with semi-coarsening in the context of smoothed aggregation and comment on the robustness of the proposed method.