Mathematical and Numerical Analysis of Some FSI Problems - Fluid-Structure Interaction and Biomedical Applications - page 42

forces H to conserve an energy estimate in the approximation process. This term
will vanish in the limit.
Furthermore, we overcome the difficulties of solenoidal spaces by means of the
artificial compressibility. We approximate the continuity equation similarly as in
[ 66 ] with

u " @p "

@t p "

C div u " D 0 in ı .t/;

@p "
@n D 0; on @ ı .t/; " > 0:

(1.99)

By letting " ! 0 we show that u " ! u ,where u is the weak solution of ( 1.73 ). For
fixed ", due to the lack of solenoidal property for velocity, we have the additional
term f

2 u i div u in momentum equation, which can be included into the convective
term, see ( 1.106 ). With both these approximation strategies we will avoid the added
mass effect for this approximation step.
The approximated problem is defined on a moving domain depending on function
h D R 0 Cı. Now we reformulate it to a fixed rectangular domain. This step requires
that the deformation h is regular enough. Let us set

u .t; x; y/ def

O D u .t; x;h.t; x/y/

p.t; x; y/ def

D p.t; x;h.t; x/y/

.t; x/ def D @ t .t; x/ (1.100)

for y 2 D Df.x; y/I 0<Ox<L;0<Oy<1g, 0<t<T.

We define the following space
V ǚ w 2 W 1;p .D/ W w 1 D 0 on S w ; w 2 D 0 on S in [ S out [ S c ;
S w Df.x;1/ W 0< Ox<Lg; in Df.0; y/ W 0< Oy<1g;

S out Df.L; y/ W 0< Oy<1g; c Df.x;0/ W 0< Ox<Lg: (1.101)

Let us introduce the following notations that correspond to the change of variables
in the problem: for the divergence equation

y
h @ x h@ y b

1
h @ y b

def
D @ x b

div h u u 1 u 1 C u 2 ;

for the bilinear form involving the pressure

Z

h @ 1 Oq@ x C @ 2 q y@ x h @ 1 q @ y ; a h .q;/ D (1.102)

D