Using high-order transport theorems for implicitly defined moving curves to perform quadrature on planar domains

The numerical integration over a planar domain that is cut by an implicitly defined boundary curve is an important problem that arises, for example, in unfitted finite element methods and in isogeometric analysis on trimmed computational domains. In this paper, we introduce a a very general version of the transport theorem for moving domains defined by implicitly defined curves and use it to establish an efficient and accurate quadrature rule for this class of domains. In numerical experiments it is shown that the method achieves high orders of convergence. Our approach is suited for high-order geometrically unfitted finite element methods as well as for high-order trimmed isogeometric analysis.