# Robust and efficient solvers for fluid-structure-interaction and high order finite element equations

## Dr. Huidong Yang

**Jan. 27, 2015, 2:30 p.m. S2 059**

In the first part of this talk, we construct robust and efficient preconditioner Krylov subspace solvers for the monolithic linear system of algebraic equations arising from the finite element discretitzation and Newton's liniarization of the fully coupled fluid-structure interaction system of Partial Differential Equations in the Arbitrary Lagrangian--Eulerian formulation. We admit nonlinear hyperelastic material in the solid model and cover a large range of flows, e.g,, water, blood, and air with highly varying density. The robust preconditioners are constructed in form of $\hat{L}\hat{D}\hat{U}$, where $\hat{L}$, $\hat{D}$ and $\hat{U}$ are proper approximations to the matrices $L$, $D$ and $U$ in the $LDU$ factorization of the fully coupled system matrix, 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. The numerical studies presented impressively demonstrate the robustness and the efficiency of the preconditioners proposed in the talk.

In the second part, we will present recently developed AMG solvers and AMG preconditioned Krylov subspace methods for the linear system of algebraic equations arising from high order finite element discretitzation of second order partitial differential equations.