Mathematical Physics - Volume II - Numerical Methods

3.4 Finite element interpolation

63

δΩ h

δΩ

δΩ

Figure 3.9: Discretization of two-dimensional problem.

It was shown that for one-dimensional problems, it is possible to represent the state variable of a unit length element in linear form: u = u 0 + u 1 ξ , Or, alternatively, in quadratic form: u = u 0 + u 1 ξ + u 2 ξ 2 . In analogous manner, we can introduce the canon triangle for a two-dimensional domain.

η

1

3

1

2

1

ξ

0

Figure 3.10: Triangular finite element.

Similar to the one-dimensional problem, wherein the unit canon length is projected into a real arbitrary length, the unit right triangle for the two-dimensional case is projected to a real triangular element of arbitrary shape and size. In addition to being simple to calculate, the triangular finite element is introduced for the purpose of natural agreement between the number of nodes and the order of the polynomial which represents the approximate solution. Namely, a two-dimensional space linear equation has the following form: v h = a 1 + a 2 ξ + a 3 η (3.64) with three constants ( a i , i = 1 , 2 , 3). In order to determine these constants, three independent values v h are necessary, which implies that the finite element needs three nodes. Furthermore, for two