Robust and efficient monolithic fluid-structure-interaction solvers

We construct new robust and efficient preconditioned generalized minimal residual solvers for the monolithic linear systems of algebraic equations arising from the finite element discretization and Newton's linearization of the fully coupled fluid–structure interaction system of partial differential equations in the arbitrary Lagrangian–Eulerian formulation. We admit both linear elastic and nonlinear hyperelastic materials in the solid model and cover a large range of flows, for example, water, blood, and air, with highly varying density. The preconditioner is constructed in form of urn:x-wiley:nme:media:nme5214:nme5214-math-0001, where urn:x-wiley:nme:media:nme5214:nme5214-math-0002, urn:x-wiley:nme:media:nme5214:nme5214-math-0003, and urn:x-wiley:nme:media:nme5214:nme5214-math-0004 are proper approximations to the matrices L, D, and U in the LDU block factorization of the fully coupled system matrix, respectively. The inverse of the corresponding Schur complement is approximated by applying a few cycles of a special class of algebraic multigrid methods to the perturbed fluid sub-problem, which is obtained by modifying corresponding entries in the original fluid matrix with an explicitly constructed approximation to the exact perturbation coming from the sparse matrix–matrix multiplications. The numerical studies presented impressively demonstrate the robustness and the efficiency of the preconditioner proposed in the paper