Partial Differential Equations - Modelling and ... - ResearchGate

12 V. Girault and M.F. Wheeler
[ ∑
|||K 1 2 ∇v|||L 2 (E h ) = ‖K 1 2 ∇v‖
2
L 2 (E)
E∈E h
and norm (for which it is a Hilbert space)
|||v||| H1 (E h ) =
] 1
2
(
)
‖v‖ 2 L 2 (Ω) + |||K 1 1
2 ∇v|||
2 2
L2 (E h ) .
In view of (9), we define the jump bilinear form
, (28)
∑ σ e
J 0 (u, v) =
[u] e [v] e dσ, (29)
h e
e∈Γ h ∪Γ
∫e
h,D
where h e denotes the diameter of e, andeachσ e is a suitable non-negative
parameter. It is convenient to define also the mesh-dependent semi-norm
(
)
[|v|] H1 (E h ) = |||K 1 1
2 ∇v|||
2
L2 (E h ) + J 2
0(v, v) . (30)
Now, we choose an integer k ≥ 1 and we discretize H 1 (E h ) with the finite
element space
X h = {v ∈ L 2 (Ω) :∀E ∈E h , v| E ∈ P k (E)}. (31)
It is possible to let k vary from one element to the next, but for simplicity we
keep the same k. Then, keeping in mind (10), we discretize (24)–(26) by the
following discrete system: Find p h ∈ X h such that for all q h ∈ X h ,
∑
K∇p h ·∇q h dx
E∈E h
∫E
−
∫
=
∑
e∈Γ h ∪Γ h,D
∫e
Ω
(
{K∇ph · n e } e [q h ] e + ε{K∇q h · n e } e [p h ] e
)
dσ + J0 (p h ,q h )
∫
fq h dx + g 2 q h dσ − ε
Γ N
+ ∑
σ e
h e
e∈Γ
∫e
h,D
∑
e∈Γ h,D
∫e
g 1 (K∇q h · n Ω ) dσ
g 1 q h dσ, (32)
with ε = 1 for SIPG, ε = 0 for IIPG and ε = −1 for NIPG and OBB-DG;
and for each e, σ e = 1 for NIPG, σ e = 0 for OBB-DG and again σ e is a well
chosen positive parameter for IIPG and SIPG.
Remark 3. Let E be an element of E h with no edge (or face) e on ∂Ω. Taking
q h = χ E , the characteristic function of E in (32), we easily derive the discrete
mass balance relation where n E denotes the unit normal exterior to E:
− ∑
e∈∂E
∫
{K∇p h }·n E dσ + ∑
e
e∈∂E
σ e
(p
h e
∫e
int
h − p ext
h ) dσ =
∫
E
fdx.