Mathematical Physics - Volume II - Numerical Methods

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

66

These linear functions are represented in Figure 3.11, and the following applies to them: ψ e i ( ξ j , η j ) = 1 , i = j 0 , i̸ = j . (3.77) In addition, we should mention that expression (3.75) can be written in condensed form as: v e h ( ξ , η ) = v i ψ e i ( ξ , η ) , (3.78) by using the aforementioned adding conventions along the same (repeated) indices. Again, observe that the expression (3.78) is more general, since it can be used for elements with an arbitrary number of interpolation functions, ψ e i , i = 1 , 2 , . . . , n . We will now determine “global” φ i ( ξ , η ) , which are obtained by “arranging” shape functions in a “pyramid”, fig. 3.11. Cartesian coordinates in this case is represented in the same way as the state variables vi are represented using interpolation functions ψ i (3.78):

x ( ξ , η ) = x i ψ i ( ξ , η ) , y ( ξ , η ) = y i ψ i ( ξ , η ) ,

(3.79)

where x i and y i are the Cartesian coordinates for the i -th node.

ψ ξ, η ( ) 1 3

1

ξ 1

Ω 1

2

3

η 1

ψ ξ, η ( ) 5 2

ξ 5

ψ ξ, η ( ) 2 1

1

η 5

Ω 5

3

2

2

ξ 2

Ω 2

1

3

η 2

1

2

Ω 4

ψ ξ,η ( ) 4 2

1

ψ ξ,η ( ) 3 1

Ω 3

2

3

3

η 4

ξ 3

η 3

φ ξ, η ( ) i

Ω 1

Ω 5

Ω 2

6

Ω 4

Ω 3

Figure 3.12: Global base functions.