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

## Dr. Huidong Yang

Jan. 27, 2015, 3:30 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.