Zienkiewicz O.C., Taylor R.L. Vol. 3. The finite - tiera.ru

294 Appendix B
As usual the domain ˆ…0; L† is partitioned into a collection of N elements
(intervals) e ˆ…xe ÿ 1; xe†; e ˆ 1; 2; ...; m. In the present case, we consider the special
weak form of Eqs (B.1) and (B.2) de®ned on this mesh by
X m
e ˆ 1
… xe
x e ÿ 1
‡ k dv
dx
k d dv d…uv†
ÿ
dx dx dx
ˆ Xm
e ˆ 1
dx ‡ Xm
e ˆ 1
k dv
dx
…L†ÿv k d
dx …L†‡v ÿ u…0†
… xe
x e ÿ 1
‰ Šÿ k d
dx ‰vŠ …x e†
fv dx ‡ k dv
…L† ‡ v…0†g ÿ uv…0† …B:3†
dx
for arbitrary weight functions v. Here h:i denotes (¯ux) averages
and ‰:Š denote jumps
hkv 0 i…x e†ˆ kv0 …x e ‡†‡kv 0 …x e ÿ†
2
…B:4†
‰ Š…x e†ˆ …x e ‡†ÿ …x e ÿ† …B:5†
it being understood that xe ˆ lim " ! 0…xe "†, v 0 ˆ dv=dx etc.
The particular structure of the weak statement in Eq. (B.3) is signi®cant. We make
the following observations concerning it:
1. If ˆ …x† is the exact solution of Eqs (B.1) and (B.2), then it is also the (one and only)
solution of Eq. (B.3); i.e. Eqs (B.1) and (B.2) imply the problem given by Eq. (B.3).
2. The solution of Eqs (B.1) and (B.2) satis®es Eq. (B.3) because is continuous and
the ¯uxes k d =dx are continuous:
‰ Š…xe†ˆ0 and k du
dx …xe†ˆ0 …B:6†
3. The Dirichlet boundary conditions (an in¯ow condition) enter the weak form on
the left-hand side, an uncommon property, but one that permits discontinuous
weight functions at relevant boundaries.
4. The signs of the second term on the left side … P efhkv 0 ‰ Ši ÿ hk 0 i‰vŠg† can be
changed without a€ecting the equivalence of Eq. (B.3) and Eqs (B.1) and (B.2),
but the particular choice of signs indicated turns out to be crucial to the stability
of the discontinuous Galerkin method (DGM).
5. We can consider the conditions of continuity of the solution and of the ¯uxes at
interelement boundaries, conditions (B.6), as constraints on the true solution.
Had we used Lagrange multipliers to enforce these constraints then, instead of
the second sum on the left-hand side of Eq. (B.3), we would have terms like
X m
e ˆ 1
f ‰ Š‡ hk d dxig…x e† …B:7†
where and are the multipliers. A simple calculation shows that the multipliers
can be identi®ed as average ¯uxes and interface jumps:
ˆ k dv
dx …xe†; ˆÿ‰vŠ…xe† …B:8†