Exact CutFEM polynomial integration

The implementation of discontinuous functions occurs in many of today's state of the art partial differential equation solvers. In finite element methods this poses an inherent difficulty: there are no quadrature rules readily available when integrating functions whose discontinuity falls in the interior of the element. Many approaches to this issue have been developed in recent years, among them is the equivalent polynomial technique. This method replaces the discontinuous function with a polynomial, potentially allowing for the integration to occur over the entire domain, rather than integrating over complex subdomains. Although eliminating the issues involved with discontinuous function integration, the equivalent polynomial tactic introduces its own set of problems. In particular, either adaptivity is required to capture the discontinuity or error is introduced when regularization of the discontinuous function is implemented. In the current work, we eliminate both of these issues. The results of this work provide exact algebraic expressions for subdomain and interface polynomial integration, where the interface represents the boundary of the cut domain. We also provide algorithms for the implementation of these expressions for standard finite element shapes in one, two, and three dimensions, along with a hypercube of arbitrary dimension, for linear discontinuities. Discontinuities defined by more general curves are analyzed and closed-form algebraic expressions are given. Practical implementation considerations are provided with a considerable level of detail.