Mathematical Physics - Volume II - Numerical Methods

Chapter 2. Finite element method

42

g z ( )

1.0

g z ( ) h

0.75

0.50

0.25

0

Cvorovi ^

z

3

5

1

4

2

z = 0

z = 1

Figure 2.5: Quadratic interpolation function with two elements.

In order to evaluate the error, let us notice that, max | g ′′ ( z ) | ≤ π 2 hence: | g ( z ) − g h ( z ) | ≤ Ch 3 , gde je C = π 2 / 48 .

(2.78)

2.3 Finite element approximation

We can now divide the domain Ω into a certain number of subdomains Ω e , with lengths h e ( ∑ h e = l ) , which will be referred to as finite elements , as seen in Fig. 2.5, for example. Let us assume a concentrated source in f , shown in Fig. ?? , located at z = ¯ z ( = z 2 in Fig. ?? ). Then flux P = ku ′ has a jump of [ | P | ] = ˆ f in ¯ z . On the other hand, element shape functions have continuous derivatives and cannot include such jumps. Hence, the finite element mesh needs to be constructed in a way that ensures all discontinuities (including jumps) are located at the nodes. In this case, terms such as ˆ f v h ( ¯ z ) , which represents a jump, are not a part of local equations that describe the approximate behaviour within the elements. Let us now consider the selection of shape functions ψ e i . Theoretical considerations enable the application of shape functions of any given order, with higher order functions providing more accurate solutions. However, due to practical reasons, only linear or quadratic elements are used, in order to avoid complications during finite element formulation.

2.4 Determining of finite element matrices and finite element sys tem matrices

For any given finite element (subdomain) Ω i between nodes S 1 and S 2 , the following holds: Z S 2 S 1 ku ′ v ′ d z = Z S 2 S 1 ¯ f v d z + P ( S 1 ) v ( S 1 ) − P ( S 2 ) v ( S 2 ) , (2.79) where P ( S i ) is the flux in nodes S 1 and S 2 , and represents the natural boundary condition.