  Standard finite element methods employ an element-wise assembly strategy. The
element's contribution to the system matrix is formed by a loop over quadrature
points. This concept is also used in fictitious domain methods, which perform
simulations on a simple tensor-product background mesh cut by a boundary
representation that defines the domain of interest.
  Considering such $d$-dimensional background meshes based on splines of degree
$p$ with maximal smoothness, $C^{p-1}$, the cost of setting up the system
matrix is $\mathcal{O}\left(p^{3d}\right)$ per degree of freedom. Alternative
assembly and formation techniques can significantly reduce this cost. In
particular, the combination of (1) sum factorization, (2) weighted quadrature,
and (3) row-based assembly yields a cost of $\mathcal{O}\left(p^{d+1}\right)$
for non-cut background meshes. However, applying this fast approach to cut
background meshes is an open challenge since they do not have a tensor-product
structure.
  This work presents techniques that allow the treatment of cut background
meshes and thus the application of fast formation and assembly to fictitious
domain methods. First, a discontinuous version of weighted quadrature is
presented, which introduces a discontinuity into a cut test function's support.
The cut region can be treated separately from the non-cut counterpart; the
latter can be assembled by the fast concepts. A three-dimensional example
investigates the accuracy and efficiency of the proposed concept and
demonstrates its speed-up compared to conventional formation and assembly.
