Introduction to Computational Linear Algebra

222 Introduction to Computational Linear Algebra

Such extension is made possible, using the Calderon-Zygmund theory, which

proves such extension with the property:

||u

0

||

1,Ω

≤ K||g||

∞,Γ

D

.

Let:

H

1

D

(Ω) = {ϕ ∈ H

1

(Ω)|ϕ = 0 on Γ

D

}.

Thus, one has:

U

ad

= u

0

+ H

1

D

(Ω).

(simply written as g + H

1

D

(Ω)). Note that the expressions in (7.46) are all well

defined, specifically in:

u ∈ U

ad

:

Z

Ω

a(x, y)∇u.∇vdxdy =

Z

Ω

f(x, y)v(x, y)dxdy + ...

... +

Z

Γ

N

h(s)v(s)ds, ∀v ∈ H

1

D

(Ω),

one requires the assumptions: f ∈ L

2

(Ω), a ∈ C(Ω), h ∈ L

2

(Γ

N

).

If A(u, v) =

R

Ω

a(x, y)∇u.∇vdxdy and F (v) =

R

Ω

f(x, y)v(x, y)dxdy +

R

Γ

N

h(s)v(s)ds, then the equivalent variational form is given by:

u ∈ {u

0

} + H

1

D

(Ω) : ...