Nonconforming and Other Methods - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 123

3.1 Definition. Let m ≥ 1. Given u ∈ L 2 () , define the norm

(u, v) 0 ,
v m,

u − m, : = sup

v ∈ H 0

.

()

We define H − m () to be the completion of L 2 () w.r.t.

· − m, .
For the Sobolev spaces built on L 2 () , we identify the dual space of H 0 ()
with H − m () . Moreover, by the definition of H − m , there is a dual pairing u, v

for all u ∈ H − m , v ∈ H 0

, i.e. u, v is a bilinear form, and

whenever u ∈ L 2 (), v ∈ H 0 ().

u, v = (u, v) 0 , ,

Clearly,

... ⊃ H − 2 () ⊃ H − 1 () ⊃ L 2 () ⊃ H 0 () ⊃ H 0 () ⊃ ...
···≤ u − 2 , ≤ u − 1 , ≤ u 0 , ≤ u 1 , ≤ u 2 , ≤···

H − m was defined to be the dual space of H 0 and not of H m . Thus we obtain an
improvement of II.2.9 only for Dirichlet problems.

3.2 Remark. Let a be an H 0 -elliptic bilinear form. Then with the notation of the
proof of the Existence Theorem II.2.9, we have

α − 1 u m ≤ f − m . ( 3 . 4 )

Proof. By Definition 3.1, (u, v) 0 ≤ u − m v m . Substituting v = u in the weak

equation gives

2
m

α u ≤ a(u, u) = (f, u) 0 ≤ f − m u m ,

and the assertion follows after dividing by u m .

This asserts that the Dirichlet problem is H m -regular in the sense of Defini-
tion II.7.1.

3.3 Remark. Let V ⊂ U be Hilbert spaces, and suppose the imbedding V → U
is continuous and dense. In addition, suppose we identify U with U via the Riesz
representation. Then

V ⊂ U ⊂ V

is called a Gelfand triple . We have already encountered the following Gelfand
triples:

H m () ⊂ L 2 () ⊂ H m () ,
H 0 () ⊂ L 2 () ⊂ H − m ().

and