  An interface preserving moving mesh algorithm in two or higher dimensions is
presented. It resolves a moving $(d-1)$-dimensional manifold directly within
the $d$-dimensional mesh, which means that the interface is represented by a
subset of moving mesh cell-surfaces. The underlying mesh is a conforming
simplicial partition that fulfills the Delaunay property. The local remeshing
algorithms allow for strong interface deformations. We give a proof that the
given algorithms preserve the interface after interface deformation and
remeshing steps. Originating from various numerical methods, data is attached
cell-wise to the mesh. After each remeshing operation the interface preserving
moving mesh retains valid data by projecting the data to the new mesh
cells.\newline An open source implementation of the moving mesh algorithm is
available at [1].
