A note on robust preconditioners for monolithic fluid-structure interaction systems of finite element equations

In this note, we consider preconditioned Krylov subspace methods for discrete fluid-structure interaction problems with a nonlinear hyperelastic material model and covering a large range of flows, e.g, water, blood, and air with highly varying density. Based on the complete $LDU$ factorization of the coupled system matrix, the preconditioner is constructed in form of $\widehat{L}\widehat{D}\widehat{U}$, where $\widehat{L}$, $\widehat{D}$ and $\widehat{U}$ are proper approximations to $L$, $D$ and $U$, respectively. The inverse of the corresponding Schur complement is approximated by applying one cycle of a special class of algebraic multigrid methods to the perturbed fluid sub-problem, that is obtained by modifying corresponding entries in the original fluid matrix with an explicitly constructed approximation of the exact perturbation coming from the sparse matrix-matrix multiplications.