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Discontinuous Galerkin Methods 19
t h (u h ; v h ,w h )= ∑ (u h ·∇v h +
E∈E
∫E
1 )
2 (div u h)v h w h dx
h
− ∑ ∫
{u h }·n E (vh
int −vh
ext )wh int dσ− 1 ∑
∫
[u h ] e·n e {v h w h } e dσ.
E∈E
(∂E) −
2
h e∈Γ h ∪∂Ω
e
(55)
This form is positive in the following sense (cf. [GRW05]): for all v h ∈ Y h
t h (u h ; v h ,v h )= 1 ∑
‖|{u h }·n E | 1 2 (v
int
h
2
E∈E h
− vh
ext )‖ 2 L 2 ((∂E) −\∂Ω)
+ ‖|u h · n Ω | 1 2 vh ‖ 2 L 2 ((∂Ω) −) , (56)
where
(∂Ω) − = {x ∈ ∂Ω : u h · n Ω (x) < 0}.
Therefore, if the penalty parameters σ e are chosen as in Section 3, we see that
system (54) has a solution t h in Y h , unique up to an additive constant. In
particular, this means that (54) is compatible with (51)–(52) and this is an
important property, cf. [DSW04].
However, proving a priori error estimates is more delicate, considering
that u h proceeds from a previous computation. If the error in computing u h
is measured in the norm (42), then the contribution of t h (u h ; c h ,v h )tothe
error is estimated as in the Navier–Stokes equations. This requires discrete
Sobolev inequalities, and as mentioned in Remark 5, this does not seem to
be possible for OBB-DG schemes. On the other hand, for IIPG, SIPG and
NIPG, the analysis in [GRW05] carries over here and yields, when u and c
are sufficiently smooth:
where k is the exponent in (50).
[|c h − c|] H1 (E h ) =O(h min(k,l) ),
Remark 7. Let E be an element as in Remark 3. Taking v h = χ E in (54), we
obtain the discrete mass balance relation:
− ∑
e∈∂E
∫
{K∇c h }·n E dσ + ∑
e
+ 1 2
e∈∂E
( ∫
(div u h )c h dx − 1
E
2
+ ∑
e∈(∂E) −
∫e
σ e
(c
h e
∫e
int
h
∫
∑
e∈∂E
(u int
h
e
− c ext
h ) dσ
− u ext
h
|{u h }·n E |(c int
h
) · n E c int dσ
h
)
∫
− c ext
h ) dσ = fdx.
E