Mathematical Physics - Volume II - Numerical Methods

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

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Let us now apply the conservation law to a material part ω , located around point P 0 , wherein all properties are smooth. If we donate a source per unit area as f , we obtain: Z ∂ω σ · n d s = Z ω f d x d y . (3.44)

By applying the divergence theorem, we obtain: Z ω

( ∇ · σ − f ) d x d y = 0

(3.45)

for all subregions ω in Ω . Since ω is an arbitrary region in which ∇ · σ and f are smooth, subintegral function in (3.45) is equal to zero for all points within ω . ∇ · σ ( x , y ) − f ( x , y ) = 0 , (3.46) which represents a local conservation law.

δ

2

2

δ

1

k k x y = ( , ) 2

n

ω

Γ

δω

P O

y

n^

P i

1

k k x y = ( , ) 1

s

x x s = ( ) y y s = ( )

δ :

x

Figure 3.5: Two-dimensional domain with an interface.

Conservation law changes its form on inter-surfaces and boundaries, as can be shown by the following analysis. If we assume that body ¯ Ω is made of two different materials, one in subregion Ω 1 and the other in Ω 2 , Fig. 3.5, we can conclude that the material modulus k is defined via smooth function (i.e. by constants) k 1 and k 2 . The curve which defines the boundary between Ω 1 and Ω 2 , will be denoted by Γ , whereas ∂ Ω 1 and ∂ Ω 2 denote parts of ∂ Ω , i.e. boundaries for which boundary conditions are defined. These boundary conditions include u = ˆ u at ∂ Ω 1 (essential boundary condition), and the natural boundary conditions at ∂ Ω 2 , which will be defined at a later stage.