Elliptic Partial Differential Equations - Lecture Notes | CSI 701, Study notes of Computer Science

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Elliptic PDE’s

Finite element methods

Finite difference methods

Discretization

-^

The solution of PDE’s are continuous functions of theindependent variables

-^

In order to represent continuous functions on a computer(an inherently discrete device) the computational domainas well as the equation operators must be discretized

-^

Usually, temporal and spatial discretization areperformed independently

-^

Functions and operators are discretized in space using computational grids

Grid examples

Body-conforming,curvilinear structured gridNon-body conforming,Cartesian grid (structured)Conforming elements

Body conforming unstructured grid

Multi-block grid Non-conforming elements

Discretization techniques

-^

Finite difference methods

-^

Finite element methods

-^

Finite volume methods

-^

Particle methods

-^

Meshless or finite points methods

Laplace’s equation

∫

∫^

∫

∫ =Ω

form

weak

on

conditions

boundary

natural

parts

by

n

integratio

statement

residual

weighted

in

equations

Laplace'

2

2

d

W

n

d

W

dn

W

d

W

φ

φ

φ

φ

φ

φ

Finite element approximation

n n

K

d

N

N

d

W

n

i N W

(x) N x

j ij

j j

i i

j j

×

∫

∫

system

global

0

form

matrix

method

Galerkin

ion

approximat

element

finite

Linear triangle: shape functions

-^

Shape functions:

-^

Area coordinates:

C B A

i i

A C

A B A

N^

x x x

x x

x x x x x x

η ξ

η ξ

η

ξ

• • − − = = ⇒

−

−

=

)

(^1) (

)

( )

(

η ζ

ξ ζ

η ξ

ζ

=

=

− − = =

3 3

2 2

1 1

1

N N N

Linear triangle: shape functions at a point

-^

The shape functions of a linear triangle can be found atthe location of a point

x

as follows: i

−

i i i

c b a

c b a

c b a

c b a

c b a

c b a

i i i

C

B

A

i

x y z

z

z

z

y

y

y

x

x

x

z

z

z

y

y

y

x

x

x

x y z

1

(^1) ( ζ ξ^ η

ζ ξ η η

ξ

η ξ^

x

x

x

x

Boundary conditions

points

boundary for

set and

system from

equation

delete

: 2

Option

RHS

in the

and

diagonal in 1 :

equations of

system

modify

: 1

Option

:

conditions

boundary

Dirichlet

n

formulatio

in the for

accounted

already

0

:

conditions

boundary

Natural

0

0

0

φ φ

φ

φ φ φ

=

=

= ⋅ ∇^

n

Remark: Shape functions continuity

-^

Given:

-^

WRM:–

N

j^ must have defined 2

nd^

order derivatives

⇒

must be at least C

1 continuous across elements

-^

W

i^ can be the

δ^ function

-^

Integrating by parts:– Order of max derivative reduced

⇒

wider space of trial functions

-^

jN must have defined 1

st^ order derivatives

⇒

must be at least C

0 continuous across elements

-^

W

i^ cannot be the

δ^ function

on

u

in

u

2

∫ Ω

j

j

i^

u d N

W

ˆ^

−^ ∫ Ω

j

j

i^

u

d

N

W

Example: Poisson’s equation

j ij

j ij

j j i j j i

j j i j j i

f M u K

d f N N u d N N

d f N W d u N W

f u =

Ω

= Ω

∇⋅ ∇

Ω

= Ω

∇⋅

∇

= ∇−

∫

∫

∫

∫

Ω

Ω

:

system

matrix

ˆ

:

method

Galerkin

ˆ

:

WRM

:

equations

Poisson'

2

Mesh

-^

Regular triangular mesh – assemble elementcontributions to produce the equation for a typical interiorpoint (point 5) for the Poisson operator:

-^

Connectivity:

f

u^

=

−∇

(^2)

Element

Node

Node 2

Node 3

1

1

5

4

2

2

5

1

3

2

6

5

4

2

3

6

5

4

8

7

6

4

5

8

7

5

9

8

8

5

6

9

Contributions of element 1

-^

Shape function derivatives:

-^

LHS contribution

-^

RHS contribution

⎤ ⎥ ⎥ ⎥⎦ ⎡−⎢ ⎢ ⎢⎣ = ⎤ ⎥ ⎥ ⎥ ⎥⎦ ⎡ ⎢ ⎢ ⎢ ⎢⎣ ⎤ ⎥ ⎥ ⎥⎦ ⎡ ⎢ ⎢ ⎢−⎣ = ⎤ ⎥ ⎥ ⎥ ⎥⎦ ⎡ ⎢ ⎢ ⎢ ⎢⎣

h h h

N N N

h h h

N N N

y A B C

x A B C

0 1

, 0 1

2 ,

2 ,

⎤ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⋅⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎥⎦

⎡ ⎢ ⎢ ⎢⎣

− −

− −

⎤ ⎥ =⎥ ⎥ ⎦ ⎡ ⎢ ⋅⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎥ ⎥⎦

⎡ ⎢ ⎢ ⎢ ⎢⎣

−

−

− −

= ⋅

1 5 4

1 5 4 2

2

2

2

2

2

2 2

1 1

ˆ ˆ ˆ 2 1 1

1 1 0

1 0 1 1 2 ˆ ˆ ˆ

2

0

0

(^12)

u u u

u u u

h

h

h

h

h

h

h h

u K

⎢⎣^

⎢⎣^

4

5 1

4

f

f f

f^

⋅^

4 5

1

4 5 1 2

1 5

2

1 1

f f

f

f f f

h

f f

h f M

Contributions of element 2

-^

Shape function derivatives:

-^

LHS contribution

-^

RHS contribution

⎤ ⎥ ⎥ ⎥⎦ ⎡ ⎢−⎢ ⎢⎣ = ⎤ ⎥ ⎥ ⎥ ⎥⎦ ⎡ ⎢ ⎢ ⎢ ⎢⎣ ⎤ ⎥ ⎥ ⎥⎦ ⎡−⎢ ⎢ ⎢⎣ = ⎤ ⎥ ⎥ ⎥ ⎥⎦ ⎡ ⎢ ⎢ ⎢ ⎢⎣

h h h

N N N

h h h

N N N

y A B C

x A^ B C

0 1

, 0 1

2 ,

2 ,

⎤ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⋅⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎥⎦

⎡ ⎢ ⎢ ⎢⎣

−

−

−

−

⎤ ⎥ =⎥ ⎥ ⎦ ⎡ ⎢ ⋅⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎥ ⎥⎦

⎡ ⎢ ⎢ ⎢ ⎢⎣

−

−

−

−

= ⋅

1 2 5

1 2 5

2 2

2

2

2

2

2 2

2 2

ˆ ˆ ˆ 1 1 0

1 2 1

0 1 1 1 2 ˆ ˆ ˆ

0

2

0

(^12)

u u u

u u u

h h

h

h

h

h

h h

u K

⎢⎣^

⎢⎣^

5

2 1

5

f

f f

f^

⋅^

5 2

1

5 2 1

2

1 2

2

1 1

f f

f

f f f

h

f f

h f M