Mathematical Physics - Volume II - Numerical Methods

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

80

Interpolation function gradient can be written as:

∂ψ i ∂ ξ α g

α , α = 1 , 2 ,

∇ ψ i ( ξ , η ) = u i

(3.137)

where ξ 1 ≡ ξ and ξ 2 ≡ η , and g α , is the contravariant base vector of coordinate systems for element ξ α . Hereinafter we will denote partial derivatives of interpolation functions in their shortened form: ψ i α = ∂ψ i ∂ ξ α . (3.138) By replacing interpolation function gradients obtained in this manner into (3.134)-(3.136), and taking into account that the element surface is: d x d y = J d ξ d η , (3.139) we obtain: k e i j = Z Ω e ψ e i α kg αβ ψ e j β J d ξ d η . (3.140) where the following applies, formally: g αβ = g α · g β . (3.141) In a practical sense, g αβ is calculated by inverting the expression given below: g αβ = δ kl z k α z l β , k , l = 1 , 2 (3.142) where z k α = ∂ z k ∂ ξ α , (3.143) while z 1 = x , z 2 = y (3.144) are the Cartesian coordinates. Taking into account (3.127), it is obvious that z k α = z k i ξ i α . For the observed two-dimensional case, the matrix of expressions given as g αβ is: g αβ = g 11 g 12 g 21 g 22 (3.145) where g 11 = z 1 1 2 + z 2 1 2 g 12 = g 21 = z 1 1 z 2 1 + z 2 1 z 2 2 (3.146)

g 22 = z

1 2

+ z 2

2

2

2

.

The inverse matrix is by definition: h g αβ i = g

1 | g αβ |

g 22

αβ −

g 11

− g 12

1

(3.147)

=

.

− g 21

It is easy to show that the determinant of matrix g αβ is:

12 = z

1 1

2

| g αβ | = g 11 g 22 − g 2

2 1 z

2 2 − z

1 2 z

= J 2 .

(3.148)