Fast formation and assembly for spline-based 3D fictitious domain methods

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.