\section{Concluding remarks}\label{sec:conc}
In this paper we have combined known shape optimization techniques, with
finite element methods on multiple overlapping meshes. The key features
of this approach has been discussed. For shape optimization problem where the
change of the domain can be parameterized as a translation or rotation, we
observe that the function space of the mesh transformation can be reduced to
constant functions. This yields a big speed-up for updating the mesh, where one
avoids deformation equations or remeshing techniques.
For problems where a part of the domain can be deformed, we have showed that by choosing a meshing describing the part that is changing, one can yield big speed-ups in the mesh deformation step.

Nevertheless, since the MultiMesh FEM is a fairly new method, it has only been explored for time-independent heat and Stokes equation. Further study of Nitsche enforcement of interface conditions is needed to be able to
provide stable finite element schemes for overlapping meshes for other equations.

In conclusion, the results reported in this paper, shows that the combination
of shape optimization holds great promise as a powerful method for avoiding
deformation equations and re-meshing. In a later paper, we will extend this
approach to time-dependent problems, with more complex state-equations.
{"config": "arxiv", "file": "1806.09821/conclusion.tex"}