Vector Curl and Gradient, Lecture Notes - Mathematics - 1, Study notes of Mathematics

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Finite Element Methods

Examples:^ •^ steady heat conduction:^ u^ =^

temperature,

k^ =^ thermal conductivity,

f^ =^ heating

Neumann boundary conditions

↔^ insulation,

Dirichlet boundary conditions

↔^ fixed boundary

temperature • k^ = 1,^ u

=^ gravitational potential (

−∇u^ =^

force),

f^ =^ mass distribution • k^ = 1,^ u

=^ fluid potential (

−∇u^ =^

velocity)

[^ k^1 k =

] k, 1 1 , (^2) kk 1 , 2 2 , 2

(or in 3-dimensions) permeability tensor arises ingroundwater flow (flow in porous media): −k∇u^ =

(Darcy) flow velocity

Always^ x

T^ kx >^0

, x^6 = 0^ ie.

k^ is positive definite

The Divergence Theorem If the vector field

w^ and the appropriate derivatives are

defined in

Ω^ and^ R

⊂^ Ω^ is sufficiently regular with

∂R

being the boundary of

R^ then ∫ ∇ · w^ dV^ = R

∫^ w^ ·^ n^ ∂R

dS

where^ n

is the outward pointing normal to

R^ on^ ∂R

(dV^ is an increment of volume,

dS^ is an increment of

surface: these will usually be omitted)

Weak solutions

(Poisson’s equation)

For appropriate

test^ functions

v, consider

u^ satisfying

∫^2 (∇u^ Ω +^ f^ )v^ = 0

Employing the derivative of a product rule:

∇ ·^ (v∇ u) =^ v∇

2 u^ +^ ∇u

· ∇v

integrating and using the Divergence Theorem

∫^ ∇ ·^ (v^ Ω

∫ ∇u) = (^2) v∇u^ Ω ∫ +^ ∇u^ Ω · ∇v

∫ so^ ∂Ω

∫∂u v = ∂n (^2) v∇u^ Ω ∫ +^ ∇u^ Ω · ∇v

eg. if^ Ω^ ⊂

(^2) R ∫^ ∇u^ · ∇ Ω

∫^ ∫ v = ∂u∂v ∂x^ ∂x Ω

∂u∂v+ ∂y^ ∂y

Important questions: what are suitable

v? Where do we

find weak solutions

u?

eg. if^ Ω^ ⊂

(^2) R ∫^ ∇u^ · ∇ Ω

∫^ ∫ v = ∂u∂v ∂x^ ∂x Ω

∂u∂v+ ∂y^ ∂y

Important questions: what are suitable

v? Where do we

find weak solutions

u?

Need appropriate spaces of functions: based on integration:

L(Ω) =^2

{^ u^ : Ω^ →

∫ R |^ u^ Ω

with the corresponding

norm^ (ie. size/magnitude of a

function)

(∫ ‖u‖ = )^1 /^22 u Ω

So we have the Cauchy-Schwarz inequality:

〈u, v〉^ =

∫^ uv^ ≤ ‖^ Ω

u‖‖v‖

Returning to the weak form:

∫^ vf^ ≤ ‖^ Ω

v‖‖f^ ‖

and so is defined if

v, f^ ∈^ L

∫^ ∇u^ · ∇^ Ω

∫ v =^ Ω ∂u∂v+ ∂x^ ∂x^

∂u∂v ∂y^ ∂y ∂u ≤ ‖ ∂x ∂v‖‖ ‖^ + ∂x

∂u∂v ‖ ‖‖^ ∂y^ ∂y

is defined if

u, u, vxy^

, v∈^ Lxy^

(Ω)^ and 2

∫∂u^ v^ ∂n^ ∂Ω

≤ ‖v‖∂

∂u‖ ‖Ω∂ ∂n Ω

is defined if

∂u v, ∈^ ∂n^

L(∂Ω)^2

This gives us the natural space in which weak solutions live:If^ f^ ∈^ L^2

(Ω)^ and^

g^ ∈^ L(∂^2

Ω)^ then weak solutions

u^ of the

above lie in the

Sobolev space

1 H(Ω)^ defined by

1 H(Ω) =

{^ u^ : Ω^ →

∂u R | u, ∂x ∂u, ∈^ L^ ∂y^

for^ Ω^ ⊂^

2 R. { 1 H(Ω) =^ u^

: Ω^ →^ R

′^ | u, u∈

} L(Ω) 2

for^ Ω^ ⊂

1 R

1 H(Ω) =

{^ u^ : Ω^ →

∂u R | u,^ ∂x ∂u∂u, ,^ ∂y^ ∂z

∈^ L(Ω)^2

for^ Ω^ ⊂^

3 R.^

Note^ Ω^ is an open set just as

(0,^ 1)^ is an open interval

We will sometimes need to refer to the closure of

Ω, and

denote this by

Ω^ ie.^ Ω = Ω

∪^ ∂Ω

The weak form is:Find^ u^ ∈ H

1 such that E^

∫^ ∇u^ · ∇^ Ω

∫ v =^ vf^ Ω

∫ +^ ∂ΩN

vgN^

for all^ v

1 ∈ H. E^0

[∫^ ∇u^ Ω

∫ · ∇v =

∫ vf + Ω

]∂u v ∂n ∂Ω