# Low-rank Linear Fluid-structure Interaction Discretizations

## TL;DR

This paper introduces low-rank iterative methods for efficient parameter-dependent fluid-structure interaction discretizations, significantly reducing computational time while maintaining high accuracy in simulations.

## Contribution

It develops low-rank GMRES and Chebyshev iteration variants with error bounds, enabling fast, accurate solutions for parameter-dependent FSI models.

## Key findings

- Truncated methods achieve residual norms below 10^{-8}
- Methods are twenty times faster than standard approaches
- Error bounds for truncation are rigorously derived

## Abstract

Fluid-structure interaction models involve parameters that describe the solid and the fluid behavior. In simulations, there often is a need to vary these parameters to examine the behavior of a fluid-structure interaction model for different solids and different fluids. For instance, a shipping company wants to know how the material, a ship's hull is made of, interacts with fluids at different Reynolds and Strouhal numbers before the building process takes place. Also, the behavior of such models for solids with different properties is considered before the prototype phase. A parameter-dependent linear fluid-structure interaction discretization provides approximations for a bundle of different parameters at one step. Such a discretization with respect to different material parameters leads to a big block-diagonal system matrix that is equivalent to a matrix equation as discussed in [KressnerTobler 2011]. The unknown is then a matrix which can be approximated using a low-rank approach that represents the iterate by a tensor. This paper discusses a low-rank GMRES variant and a truncated variant of the Chebyshev iteration. Bounds for the error resulting from the truncation operations are derived. Numerical experiments show that such truncated methods applied to parameter-dependent discretizations provide approximations with relative residual norms smaller than $10^{-8}$ within a twentieth of the time used by individual standard approaches.