Stokes flow context. Such a step-like geometry is illustrated in Fig. 3.
2.2 Finite element discretization
To establish notation, let us review some functional and discrete spaces. For simplicity, we suppose that Q is a poly-
gonal two-dimensional domain. The space L
2
¼ L
2
Q is a regular set of square integrable functions over Q, on which
·,· represents the usual inner product, associated with the norm
k·k. The space H
1
¼ H
1
Q denotes the usual Sobolev space of functions in L
2
whose first derivatives are also contained in L
2
. Let C be the space of functions continuous
on Q. Q
h
represents here a standard finite element triangula- tion of Q with h, where the quantity h ¼ max
t[Q
h
h
t
is a measure of the granularity of the triangulation. The spaces
of linear and cubic polynomials are denoted as P
1
and P
3
, respectively. Now we can define the following discrete
spaces: H
1 h
¼ {q
h
lq
h
[ C Q
, q
h
l
t
[ P
1
, ;t [ Q
h
} 4
V
1 h
g
¼ {~
v
h
l~v
h
[ C
Q
2
, ~v
h
l
t
[ P
p 1t
3 P
p 1t
, ~v
h
l
G
¼ ~ g
, ;t [ Q
h
} ð
5Þ with
P
p 1t
¼ {q
lq ¼ q
1
þ kf
t
, q
1
[ P
1
,
k [ R
, f
t
[ P
3
, f
t
l
]t
¼ ,
f
t
G
t
¼ 1}
ð 6Þ
where G
t
is the centroid of the triangle t. A function like f
t
is usually called a bubble-function.
1
In eqn 1, due to the incompressibility condition ~ =·~
u ¼ 0, the finite element discretization spaces for the velocity and
the pressure need to satisfy a compatibility condition, also called ‘inf-sup condition’ or ‘Ladyzhenskaya–Babusˇka–
Brezzi LBB condition’,
9
which is equivalent to the requirement of nonsingularity of the matrix resulting from
the discretized Stokes system, and thus guarantees, if satis- fied, the existence and uniqueness of the solution. In parti-
cular, this condition implies a higher number of velocity degrees of freedom than pressure unknowns. This condition
Fig. 2. Computational domain No. 2.
Fig. 3. Computational domain No. 3.
Adaptive finite element simulation of Stokes flow in porous media 19
is: inf
p[L 2
Q pÞ0
sup
~ u[ðH
1 Q
Þ2 ~
uÞ0
~ =·~
u ,
p k~=~uk·kpk
c . 0 7
where c is a positive constant. Among possible space dis- cretizations satisfying the LBB condition, we choose the
‘mini-element’ formulation,
1
which permits the resolution of the Stokes problem on a single grid for all the unknowns.
The basic idea of the mini-element formulation is that the pressure is discretized by polynomials of degree 1 P
1
, while the velocity is also discretized by polynomials of
degree 1, augmented by a polynomial of degree 3 bub- ble-function which vanishes on the edges of the triangle
t. With the mini-element formulation, the discretized Stokes system is written as:
Find ~
u
h
, p
h
[ V
1 h
g
3 H
1 h
such that m
~ =
~ u
h
, ~
= ~v
¹ p
h
, ~
=·~v ¼ ~
; ~v [ V
1 h
¹ q
, ~
=·~ u
h
¼ ;q [ H
1 h
8
: 8
where ~ v and q are the test functions associated with ~
u
h
and p
h
, respectively. Using elementwise integration, the ‘mini- element’ discretization leads eqn 8 to the following stiff-
ness matrix equation of ~
u
h
[ V
1 h
g
, p
h
[ H
1 h
for each t: A
t
e
t
B
t t
, x
e
t t
j
9 t
w
t t
, x
A
t
e
t
B
t t
, y
e
t t
j
9 t
w
t t
, y
B
t ,
x
w
t ,
x
B
t ,
y
w
t ,
y
B B
B B
B B
B B
B 1
C C
C C
C C
C C
C A
u
h ,
l
u
h ,
b
v
h ,
l
v
h ,
b
p
h ,
l
B B
B B
B B
B B
1 C
C C
C C
C C
C A
¼ B
B B
B B
B B
B 1
C C
C C
C C
C C
A
9 where the solution
~ u
h
, p
h
can be uniquely decomposed into its linear part
~ u
h ,
l
, p
h
at three vertices of the triangle t and its bubble part
~ u
h ,
b
, at the centoid of t here, the
subscripts l and b represent the piecewise linear and cubic interpolations, respectively; thus, for the triangle t, there
are totally 11 unknowns: 4 unknowns for each component of the velocity and 3 for the pressure. The 3 3 3 matrices
A
t
,B
t,x
,B
t;y
correspond to inner products involving linear basis function for the velocity and the pressure, the scalar
j
9 t
is the contribution to the H
1
inner product from the cubic bubble functions for the velocity, and the 3-vectors w
t,w
and w
t,y
correspond to contributions to the divergence term for bilinear basis functions 1-order derivative of the
cubic bubble functions for the velocity and linear functions for the pressure. The detailed expressions of
each bloc in eqn 9 can be found elsewhere
11
where the Stokes problem is studied in its most general form.
For further discussion, we specify here only the expression of j
9 t
: j
9 t
¼ m
h
2 1
þ h
2 2
þ h
2 3
720 ltl
10 where h
i
i ¼ 1,2,3 and ltl are the length of three edges and
the area of t, respectively. As mentioned, the bubble function turns out to be zero on
the triangle edges. We use in fact only the linear part of the velocity solution. In practice, we can eliminate formally the
bubble unknowns u
h,b
and v
h,b
from the left-hand side of eqn 9. This can be done elementwisely. With the bubble
unknowns statically condensed,
11
eqn 9 is now reduced into the following form:
A
9 t
B
9 t t
, x
A
9 t
B
9 t t
, y
B
9 t
, x
B
9 t
, y
¹ C
9 t
B B
B 1
C C
C A
u
h ,
l
v
h ,
l
p
h ,
l
B B
1 C
C A
¼ F
1
F
2
H B
B 1
C C
A 11
Interested readers are referred elsewhere
11
for the details of each bloc in eqn 11. We give here only the form of C
9 t
which is no longer zero as in eqn 9: C
9 t
¼ 1
j
9 t
w
t ,
x
w
t t
, x
þ w
t ,
y
w
t t
, y
12 In fact, the solvability of the Stokes problem is owing to the
condensation terms appearing in eqn 12.
2.3 A Stokes solver
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