Selective enrichment of moment fitting and application to cut finite elements and cells

For many computational methods, the important task of integrating discontinuous functions can turn out to be a bottleneck. In extended finite element methods as well as in fictitious domain or immersed boundary schemes or cut finite elements, the discretization of the problem of interest involves the computation of integrals with discontinuous functions. To this end, we propose an efficient moment fitting scheme applicable to high-order methods, accounting for discontinuities on curved interfaces. The integration points are pre-selected as the standard Gauss-Legendre abscissae, and the basis functions of the moment fitting are defined in such a way that solving an equation system can be circumvented, thus reducing the overhead of the proposed approach significantly. An example related to a mechanical problem of a 1D bar exhibiting a material interface serves to demonstrate the possibility of adjusting the enrichment of the basis function of the moment fitting with the trial and test function space of the Galerkin method. Due to the multiplicative or tensor product nature of the set of basis functions to be integrated, a selective approach is proposed in order not to increase the number of integration points. In this way, high precision can be achieved for a p-extension of the spatial discretization, yielding an exponential convergence of the error in energy norm without the necessity of refining the mesh. The computation of the required quadrature rule for the discontinuous integrand is carried out in an offline stage. The related overhead is relatively small and amortized during the evaluation of costly integrals such as the computation of high-order stiffness matrices. The approach should lead to a huge time saving, especially in the case of nonlinear problems where the quadrature rule can be reused many times. Two-dimensional and three-dimensional problems show that the proposed moment fitting can also be applied for the integration of high-order functions including discontinuities across curved boundaries.