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110 3. Finite Element Methods for Linear Elliptic Problems

Remark 3.17 In the existence result of Theorem 3.1, the only assumption
is that b has to be a continuous linear form in V .
For d =1and Ω = (a, b) this is also satisfied, for instance, for the special
linear form
1
δ γ (v):= v(γ)for v ∈ H (a, b),
where γ ∈ (a, b) is arbitrary but fixed, since by Lemma 3.4 the space
1
H (a, b) is continuously embedded in the space C[a, b]. Thus, for d =1
point sources (b = δ γ ) are also allowed. However, for d ≥ 2this doesnot
¯
1
hold since H (Ω) ⊂ C(Ω).
Finally, we will once again state the general assumptions under which the
variational formulation of the boundary value problem (3.12), (3.18)–(3.20)
in the space (3.30),
1
V = v ∈ H (Ω) : γ 0 (v)= 0 on Γ 3 ,
has properties that satisfy the conditions of the Lax–Milgram Theorem
(Theorem 3.1):
d
• Ω ⊂ R is a bounded Lipschitz domain.
2
∞
• k ij ,c i , ∇· c, r ∈ L (Ω) ,f ∈ L (Ω) , i, j ∈{1,... ,d}, and, if
∞
|Γ 1 ∪ Γ 2 | d−1 > 0, ν · c ∈ L (Γ 1 ∪ Γ 2 ) (i.e., (3.15)).
• There exists some constant k 0 > 0 such that in Ω, we have ξ·K(x)ξ ≥
2 d
k 0 |ξ| for all ξ ∈ R (i.e., (3.16)),
2
∞
• g j ∈ L (Γ j ) ,j =1, 2, 3,α ∈ L (Γ 2 ) (i.e., (3.22)).
• The following hold:
1
(1) r − ∇· c ≥ 0inΩ .
2
(2) ν · c ≥ 0on Γ 1 .
1
(3) α + ν · c ≥ 0on Γ 2 .
2
(4) Additionally, one of the following conditions is satisfied:
(a) |Γ 3 | d−1 > 0 .
˜
˜
(b) There exists some Ω ⊂ Ωwith |Ω| d > 0and r 0 > 0such
˜
1
that r − ∇· c ≥ r 0 on Ω.
2
˜
˜
(c) There exists some Γ 1 ⊂ Γ 1 with |Γ 1 | d−1 > 0and c 0 > 0
˜
such that ν · c ≥ c 0 on Γ 1 .
˜
˜
(d) There exists some Γ 2 ⊂ Γ 2 with |Γ 2 | d−1 > 0and α 0 > 0
˜
1
such that α + ν · c ≥ α 0 on Γ 2 .
2
1
• If |Γ 3 | > 0 , then there exists some w ∈ H (Ω) with γ 0 (w)= g 3
d−1
on Γ 3 (i.e., (3.34)).