Efficient Implementation of Finite Element Methods on Nonmatching and Overlapping Meshes in Three Dimensions

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.