Mathematical Physics - Volume II - Numerical Methods

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

56

∂ u ∂ x

∂ u ∂ y

d u d t

= ∇ u · t =

cos θ +

sin θ .

(3.37)

( ) s

n ( ) s

s

n ( ) s

s

Ω

( ) s n

( ) s

y

( ) s τ

x

a)

b)

Figure 3.3: Flux in two-dimensional domain.

The second physical quantity we are interested in is the flux σ , which is a vector field. Flux σ is represented by arrows in ¯ Ω , i.e. by vector σ ( s ) at point s on the boundary ∂ Ω , Figure 3.3. The flux passing through this boundary at point s is defined by its component:

σ n ( s ) = σ ( s ) · n ( s ) ,

(3.38)

where n ( s ) is the normal at the boundary ∂ Ω at point s , Fig. 3.3b. Tangential component τ ( s ) is given as σ ( s ) · τ ( s ) , Fig. 3.3b. Let us consider an arbitrary subregion ω , containing a point P 0 ( x 0 , y 0 ) . Shown in Fig. 3.4 is the distribution of σ n ( s ) along the boundary ∂ω , and the total flux passing through this boundary is given by: Σ ω ≡ Z ∂ω σ n ( s ) d s (3.39) If we divide Σ ω by the subregion A ω surface area, we obtain the mean value of flux σ which enters ω per unit surface. The limit value of this quotient, when ω is decreasing, while containing point P 0 is referred to as the flux divergence in point P 0 , and is denoted as div σ ( x 0 , y 0 ) . If we adopt that ω is a square subregion with P 0 , Fig. 3.4b, we obtain that Σ ω = ∆ σ x ∆ y + ∆ σ y ∆ x , hence the mean value theorem gives us:

∂σ y ∂ y

∂σ x ∂ x

div σ =

(= ∇ · σ ) .

(3.40)

+