Nonconforming and Other Methods - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 128

Problems

3.8 be a positive symmetric bilinear form satisfying the

hypotheses of Theorem 3.6. Show that a is elliptic, i.e., a(v, v) ≥ α 1 v

Let a : V × V → R

2
V

for

some α 1 > 0.

3.9 [Nitsche, private communication] Show the following converse of Lemma
3.7: Suppose that for every f ∈ V , the solution of (3.5) satisfies

= L − 1 f. lim

h 0 u h = u : →

Then

a(u h ,v h )
u h U v h V

in h inf

u h ∈

sup
v h ∈

> 0 .

U h V h

Hint: Use (3.10) and apply the principle of uniform boundedness.

3.10 Show that

2
0

for all v ∈ H 0 (), v ≤ v m v − m

2
1

for all v ∈ H 2 () ∩ H 0 (). v ≤ v 0 v 2

Hint: To prove the second relation, use the Helmholtz equation − u + u = f .

3.11 Let L be an H 1 -elliptic differential operator. In which Sobolev spaces H s ()
is the set

{ u ∈ H 1 () ; Lu = f ∈ L 2 (), f 0 ≤ 1 }

compact?

3.12 (Fredholm Alternative) Let H be a Hilbert space. Assume that the linear
mapping A : H → H can be decomposed in the form A = A 0 + K , where A 0 is
H -elliptic, and K is compact. Show that either A satisfies the inf-sup condition,
or there exists an element x ∈ H , x = 0, with Ax = 0.

d

H − 1 () and 3.13 Let ⊂ R and p ∈ L 2 () . Show that grad p ∈

grad p − 1 , ≤ d p 0 , . ( 3 . 14 )

Hint: Start with proving (3.14) for smooth functions and use Green's formula.
Complete the proof by a density argument.