Mathematical and Numerical Analysis of Some FSI Problems - Fluid-Structure Interaction and Biomedical Applications - page 69

1.4 Conclusions

Our main aim in this chapter was to discuss analytical and numerical difficulties
arising in fluid-structure interaction problems. Due to the nonlinear geometrical
coupling the so-called added mass effects appear and play a crucial role in both
analysis and numerical simulations. The applications we have in mind are, for
example, blood flow in elastic vessels, but more general applications of fluid flow in
compliant domains are governed as well. More importantly, if the coupled problem
has biological applications the fraction of fluid and structure densities is typically
close to one and the added mass effects due to decoupling fluid and structure
are more profound. We have presented several approaches to obtain existence of
global weak solutions for both Newtonian and non-Newtonian shear-dependent
fluids. Moreover, we present also main ideas and corresponding results for local
existence of strong solutions. In this chapter we concentrate on the case when a
two-dimensional fluid interacts with a viscoelastic membrane. However, many of the
results can be generalized for three-dimensional situations and purely elastic struc-
ture and we present a broad literature overview. In the second part we concentrate on
the numerical analysis and try to underline connections between analytical results
and construction of efficient and stable schemes. We present in more detail two
partitioned schemes, the semi-implicit scheme, cf. Sect. 1.3.2 and the kinematically
coupled scheme based on a suitable operator splitting technique 1.3.3 . We also study
their stability and accuracy from both theoretical and experimental point of view.
The chapter is closed with some numerical experiments that demonstrate reliability
of the presented numerical approaches.

Acknowledgements Céline Grandmont was supported by the grant ANR-08-JCJC-013-01
(M3RS project) of the French Research National Agency and by REO Project, Inria Paris
Rocquencourt, Inria, France & LJLL, UPMC Univeristy, Paris, France, Mária Lukácová-
Medvid'ová was supported by the German Science Foundation under the grant LU 1470/2-2,3,
and Šárka Necasová was supported by the Grant Agency of the Czech Republic n. P 201/11/1304
and by RVO 67985840.

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