Mathematical Physics - Volume II - Numerical Methods

Chapter 2. Finite element method

36

2.1.2 Galerkin’s method Assuming that weak (variation) formulation given by (2.26), which holds for all functions v ∈ H 1 . Function H 1 , in addition to its aforementioned properties, is a linear function space, with an infinite number of dimensions. The term linear space means that any linear combination of functions from H 1 is also a function from H 1 . Thus, if v 1 and v 2 are weight functions, then α v 1 + β v 2 is also a weight function. The term infinitely dimensional means that determining of function v in a space requires an infinite number of parameters. Let us assume that an infinite set of functions φ 1 ( z ) , φ 2 ( z ) . . . in H 1 , wherein every weight function can be expressed as a linear combination of functions φ i ( z ) .

∞ ∑ i = 1

β i φ i ( z ) ,

v ( z ) =

(2.29)

where β i are constants, and series defined by (2.29) converges "in the sense of H 1 ". Set of functions P i , which satisfy this condition, provides a base for H1, hence such functions are referred to as base functions. If we assume a finite number of series terms N , instead of an infinite amount, we obtain the approximation function of v , denoted as v N :

N ∑ i = 1

β i φ i ( z ) .

v N ( z ) =

(2.30)

1 ", if the

In the case v N is given by (2.30), then v N converges towards v "in the sense of H

following holds:

lim N → ∞ Z

l

2 d z = 0 .

( v ′ − v ′ N )

(2.31)

0

A total of N base functions { φ 1 , φ 2 . . . φ N } defines an N -dimensional subspace H 1 . Additionally, let us assume that these functions are linearly independent. In this case, we can formulate Galerkin’s method as a means of determining the approximate solution of equation (2.26) in a finite-dimensional subspace H ( N ) of space H 1 , which contains allowable functions (rather than the entire H1 space). Thus, instead of observing an infinitely-dimensional problem, we seek an approximate solution in the following form: ( N ) of space H 1 , since every function φ i , i = 1 , . . . , N is a member of H

N ∑ i = 1

( N ) ,

α i φ i ( z ) φ i ( z ) ∈ H

u N ( z ) =

(2.32)

the above form satisfies the conditions given by (2.26) in the case H 1 is replaced by H ( N ) , where α i represent approximation degrees of freedom. We can now formulate the variation base fr the approximate problem: find a function u N ∈ H ( N ) , so that the following holds: Z l 0 ku ′ N v ′ N d z = Pv N l 0 + Z l 0 f v N d z for all v N ∈ H ( N ) . (2.33) By replacing (2.30) and (2.32) into (2.33), we obtain: Z l 0 k N ∑ i = 1 β i φ i ! ′ N ∑ j = 1 α j φ j ! ′ d z = P N ∑ i = 1 ( β i φ i ) l 0 + Z l 0 f N ∑ i = 1 β i φ i d z , (2.34) In other words, we have: N ∑ i = 1 β i N ∑ 1 Z l 0 k φ ′ i φ ′ j d z α j − P φ i l 0 − Z l 0 f φ i d z ! = 0 , za ∀ β i , (2.35)