Mathematical Physics - Volume II - Numerical Methods

Chapter 3. Comparison of finite element method and finite difference method

70

where N is the global system matrix dimensions, i.e.:

E ∑ e = 1

( K e

e i + Σ

e i ) = 0 , i = 1 , · · · , N .

i j u j − F

(3.100)

It should be noticed that contributions from boundary conditions to matrix K i j and vector F i originate from term Σ e i , which can be written in following form:

E ∑ e = 1

( 0 ) i

( 1 ) i

( 2 ) i

Σ e

i = S

+ S

+ S

(3.101)

where

E ∑ e = 1 Z ∂ Ω

S ( 0 ) i

σ n φ i d s ,

(3.102)

=

e − ∂ Ω h

= Z = Z

S ( 1 ) i

σ n φ i d s

(3.103)

∂ Ω 1 h

S ( 2 ) i

σ n φ i d s .

(3.104)

∂ Ω 2 h

Symbol ∂ Ω e − ∂ Ω h denotes the segment of boundary ∂ Ω e , which does not include ∂ Ω h (i.e. the part of ∂ Ω e which is related to common element boundaries), hence term S ( 0 ) i is only defined in internal nodes.

2

5

4

1

4

1

3

1

3

3

2

2

4

Figure 3.14: Partition into inner elements.

For the purpose of a more thorough interpretation, figure 3.14, we will analyse four internal elements with a common node 1, for which the following holds:

4 ∑ e = 1 Z ∂ Ω

S ( 0 )

σ n φ i d s =

1 =

e

= Z Γ 1

[ | σ n | ] φ i d s + Z Γ 2

[ | σ n | ] φ i d s + Z Γ 3

[ | σ n | ] φ i d s + Z Γ 4

[ | σ n | ] φ i d s .