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102 3. Finite Element Methods for Linear Elliptic Problems

In the following the above steps will be described for some important special
cases.

(I) Homogeneous Dirichlet Boundary Condition
1
∂Ω= Γ 3 ,g 3 ≡ 0 ,V := H (Ω)
0
Suppose u is a solution of (3.12), (3.20); that is, in the sense of classical
¯
2
solutions let u ∈ C (Ω) ∩ C(Ω) and the differential equation (3.12) be
satisfied pointwise in Ω under the assumptions (3.13) as well as u =0
2
pointwise on ∂Ω. However, the weaker case in which u ∈ H (Ω) ∩ V and
2
the differential equation is satisfied in the sense of L (Ω), now under the
assumptions (3.14), can also be considered.
Multiplying (3.12) by v ∈ C (Ω) (in the classical case) or by v ∈ V ,
∞
0
respectively, then integrating by parts according to (3.11) and taking into
1
∞
account that v =0 on ∂Ω by virtue of the definition of C (Ω) and H (Ω),
0
0
respectively, we obtain

a(u, v):= {K∇u ·∇v + c ·∇uv + ruv} dx (3.23)
Ω

∞
= b(v):= fv dx for all v ∈ C (Ω) or v ∈ V.
0
Ω
The bilinear form a is symmetric if c vanishes (almost everywhere).
2
For f ∈ L (Ω),
b is continuous on (V, · 1 ) . (3.24)
This follows directly from the Cauchy–Schwarz inequality, since

|b(v)|≤ |f||v| dx ≤ f 0 v 0 ≤ f 0 v 1 for v ∈ V.
Ω
Further, by (3.15),
a is continuous (V, · 1 ) . (3.25)
Proof: First, we obtain

|a(u, v)|≤ {|K∇u||∇v| + |c||∇u||v| + |r||u||v|} dx .
Ω
Here |· | denotes the absolute value of a real number or the Euclidean
norm of a vector. Using also · 2 for the (associated) spectral norm, and
∞
· ∞ for the L (Ω) norm of a function, we further introduce the following
notations:
* +

C 1 := max K 2 , r ∞ < ∞ , C 2 := |c| < ∞ .

∞ ∞
By virtue of
|K(x)∇u(x)|≤ K(x) 2 |∇u(x)|,