  An accurate implicit description of geometries is enabled by the level-set
method. Level-set data is given at the nodes of a higher-order background mesh
and the interpolated zero-level sets imply boundaries of the domain or
interfaces within. The higher-order accurate integration of elements cut by the
zero-level sets is described. The proposed strategy relies on an automatic
meshing of the cut elements. Firstly, the zero-level sets are identified and
meshed by higher-order interface elements. Secondly, the cut elements are
decomposed into conforming sub-elements on the two sides of the zero-level
sets. Any quadrature rule may then be employed within the sub-elements. The
approach is described in two and three dimensions without any requirements on
the background meshes. Special attention is given to the consideration of
corners and edges of the implicit geometries.
