  In recent years, a number of finite element methods have been formulated for
the solution of partial differential equations on complex geometries based on
non-matching or overlapping meshes. Examples of such methods include the
fictitious domain method, the extended finite element method, and Nitsche's
method. In all of these methods, integrals must be computed over cut cells or
subsimplices which is challenging to implement, especially in three space
dimensions. In this note, we address the main challenges of such an
implementation and demonstrate good performance of a fully general code for
automatic detection of mesh intersections and integration over cut cells and
subsimplices. As a canonical example of an overlapping mesh method, we consider
Nitsche's method which we apply to Poisson's equation and a linear elastic
problem.
