The interest in finite element methods on non-matching meshes has increased in recent years. Theoretical advances, based on the ideas of Nitsche, have allowed for the construction of methods such as CutFEM for a wide variety of problems.
From the perspective of implementation, one major challenge is to be able to compute integrals over cells that only have a partial intersection with the domain. Often, quadrature rules for both the volume and surface are required by the method.
In this contribution we will present our work on using custom quadrature in the FEniCSx framework. We focus on domains described by level sets as well as domains from CAD, and restrict ourselves to quadrilateral and hexahedral elements.