Vector Curl and Gradient, Lecture Notes - Mathematics - 3, Study notes of Mathematics

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For boundary value problem:Find

u^ such that

−∇

2 u^ =

f^

in Ω

u^ =

gD^

on^

∂ΩD

and

∂u ∂n^

=^ g

N^

on^

∂ΩN

The weak form is:Find

u^ ∈ H

1 such that E^ ∫^ ∇^ Ω

u^ · ∇

v^ =

∫^ vf^ Ω

∫ + ∂ΩN

vgN

for all

v^ ∈ H

1. E^0

(^1) H E^ =^ {

u^ ∈ H

1 (Ω)

|^ u^

=^ g

onD ∂Ω

},D

(^1) H E^0 =^ {

v^ ∈ H

1 (Ω)

|^ v^

= 0 on

∂Ω

},D

2-dimension problem:

^

^











^ ^      

^ ^    

^ ^    !! !! !! !! !!

%%%^ &&&

+++^ ,,,

Want to construct S⊂ H^0

1 = E^0

{v^

: Ω^

→^ R

|^ v, v

, vxy

∈^ L

(Ω) 2 , v^ = 0

on^ ∂

Ω}D^

Cover (tile/tesselate) the domain

Ω^ with non-overlapping

triangles

4 k

, k^ = 1

,... , K

(a^ triangulation

) which are such

that^ •^

⋃^4 k^

=k Ω,

-^4

∩ 4`

=m ∅^ for

`^6 =

m.

-^ vertices of neighbouring triangles coincideTriangle vertices meet at

nodes

.

Surrounding any node not on

∂Ω

is aD patch

of triangles

that each have that node as a vertex.Label the nodes

j^ = 1

,... , n

, then for each

j, define a

basis function

φj^

by

-^ φ

is continuousj

-^ φ

= 0j

outside the patch around node

j

-^ φ

is a linear on each triangle ie. of the formj (^) ax + by^ +^ c

-^ φ

= 1j

at node

j^ (note the above

⇒^

φ= 0j^

at all other

nodes of the triangulation)

Define

S^0

by

S=^0

span

{φ^1

, φ^2

,... , φ

}n

clearly

{φ^1

, φ^2

,... , φ

}^ is linearly independent so it is an

basis for

S— the basis of^0

P^1 or (continuous)

piecewise

linear

functions on the triangulation. Choose also piecewise linear functions φn+

, φn

,... , φ+

n+n

in a similar way for nodes∂

j^ =

n^ + 1

,... , n

+^ n

which lie on∂

∂Ω

and seekD

u=h^

n∑^ u j=

φj j^

n + +n∂∑ j=n+

uφj^

∈ Hj

1 E

such that

∫ ∇ Ω u· ∇h^

v^ =

∫^ vf^ Ω

∫ + ∂ΩN

vgN

for all

v^ ∈

S^0

If^ A

is invertible, then solution of the linear system gives u = (u

,^ u 1 ,... , 2

u)n

T^ and hence the Galerkin approximate

solution

u=h^

n∑^ u j=

φj j^

n + +n∂∑ j=n+

uφj^

∈ Hj

1 E

Note that each

φj^

so defined satisfies

φj^

∈ H

1 : its

derivative is square integrable even though it is notcontinuous.The piecewise linear ‘tent’ functions are the ‘most locallydefined basis’ for

S: they have the smallest possible^0

support

⇒^

the matrix

A^ is

sparse

. ie. it has many zero

entries.This is very important for efficient numerical solution of thesystem of linear equations especially when

n^ is large.