Mathematical and Numerical Analysis of Some FSI Problems - Fluid-Structure Interaction and Biomedical Applications - page 29

and

Z T Z

B j u . u / j

2

"; (1.61)

0

with extended by 0 for t<0 and u and @ t extended by 0 for t<0 , and where
denotes the characteristic function of O T .
Remark 1.7. The main difference between the two lemmas is that in Lemma 1.2 ,
a uniform rate is obtained, whereas in Lemma 1.3 , we have only a convergence to
zero, as h goes to zero, uniformly in LJ 2 . Note that the dissipation of the fluid, which
induces dissipation of the structure, is crucial.

In order to prove this lemma we may follow the same lines as for the proof of
Lemma 1.2 and take advantage of the fact that @ t 2 L 2 .0;T I H s .0;L// for any s<
1=4 thanks to the equality of the velocities at the interface. Note that it is necessary
to split the space of high and low frequencies of the structure velocity. The space
of high frequencies are controlled since @ t 2 L 2 .0;T I H s .0;L//, with s < 1=4
and the low frequencies will be controlled thanks to the variational formulation
and a good choice of the test functions. Let us consider the eigenfunctions
i associated with the Laplace operator on .0;L/ with homogeneous Dirichlet

boundary conditions and satisfying the additional constraint R L

0 i D 0. They form
abasisofH 0 .0;L/ \ L 0 .0;L/. Consequently the L 2 -projection, denoted N 0 ,of
in the space generated by the N 0 first eigenvectors satisfies

k L 2 .0;L/ C s=2

k@ t @ t N 0

k@ t k H s .0;L/ ;s < 1=4;

N 0

where N 0 is the eigenvalue associated with N 0 . Then the high frequencies of @ t
are controlled in L 2 .0;T I L 2 .0;L// uniformly in LJ 2 and we have
Z T

Z L

s
2
N 0 ;s < 1=4;

C .@ t hf;N 0 .@ t hf;N 0 /.t h// 2

0 0

with hf;N 0 D N 0 and where C is independent of the structure viscosity LJ 2 .

For the low frequency part @ t N 0 , the variational formulation will be used to
obtain a uniform convergence (with respect to LJ 2 when h goes to zero) of quantities
k u .t C h/ u .t/k L tx and k@ t N 0 .t C h/ @ t N 0 .t/k L tx . The structure test function

is b D R t

t h @ t N " . The fluid test function has to be chosen carefully. We set

Z t Z t

t h R Ǜ .@ t N 0 /.s/ds;

. u R Ǜ .@ t /

D .s/ ds C

t h