Artigo sobre o método de Galerkin

1. Computational Engineering
Galerkin Method
Yijian Zhan
Ning Ma

2. Galerkin Method
 Engineering problems: differential
equations with boundary conditions.
Generally denoted as: D(U)=0; B(U)=0
 Our task: to find the function U which
satisfies the given differential equations
and boundary conditions.
 Reality: difficult, even impossible to solve
the problem analytically

3. Galerkin Method
 In practical cases we often apply
approximation.
 One of the approximation methods:
Galerkin Method, invented by Russian
mathematician Boris Grigoryevich Galerkin.

4. Galerkin Method
Related knowledge
 Inner product of functions
 Basis of a vector space of functions

5. Galerkin Method
Inner product
 Inner product of two functions in a certain
domain:
shows the inner
product of f(x) and g(x) on the interval [ a,
b ].
*One important property: orthogonality
If , f and g are orthogonal to each
other;
**If for arbitrary w(x), =0, f(x) 0
, ( ) ( )
b
a
f g f x g x dx  
, 0f g 
,w f  

6. Galerkin Method
Basis of a space
 V: a function space
 Basis of V: a set of linear independent
functions
Any function could be uniquely
written as the linear combination of the
basis:
0{ ( )}i iS x 

( )f x V
0
( ) ( )j j
j
f x c x


 

7. Galerkin Method
Weighted residual methods
 A weighted residual method uses a finite
number of functions .
 The differential equation of the problem is
D(U)=0 on the boundary B(U), for example:
on B[U]=[a,b].
where “L” is a differential operator and “f”
is a given function. We have to solve the
D.E. to obtain U.
0{ ( )}n
i ix 
( ) ( ( )) ( ) 0D U L U x f x  

8. Galerkin method
Weighted residual
 Step 1.
Introduce a “trial solution” of U:
to replace U(x)
: finite number of basis functions
: unknown coefficients
* Residual is defined as:
0
1
( ) ( ) ( )
n
j j
j
U u x x c x 

   
jc
( )j x
( ) [ ( )] [ ( )] ( )R x D u x L u x f x  

9. Galerkin Method
Weighted residual
 Step 2.
Choose “arbitrary” “weight functions” w(x),
let:
With the concepts of “inner product” and
“orthogonality”, we have:
The inner product of the weight function
and the residual is zero, which means that
the trial function partially satisfies the
problem.
So, our goal: to construct such u(x)
, ( ) , ( ) ( ){ [ ( )]} 0
b
a
w R x w D u w x D u x dx   

10. Galerkin Method
Weighted residual
 Step 3.
Galerkin weighted residual method:
choose weight function w from the basis
functions , then
These are a set of n-order linear
equations. Solve it, obtain all of the
coefficients .
j
0
1
, [ ( )] ( ){ [ ( ) ( )]} 0
nb
j j j ja
j
w R D u dx x D x c x dx   

     
jc

11. Galerkin Method
Weighted residual
 Step 4.
The “trial solution”
is the approximation solution we want.
0
1
( ) ( ) ( )
n
j j
j
u x x c x 

  

12. Galerkin Method Example
 Solve the differential equation:
with the boundary condition:
( ( )) ''( ) ( ) 2 (1 ) 0D y x y x y x x x    
(0) 0, (1) 0y y 

13. Galerkin Method Example
 Step 1.
Choose trial function:
We make n=3, and
0
1
( ) ( ) ( )
n
i i
i
y x x c x 

  
0
1
2 2
2
3 3
3
0,
( 1),
( 1)
( 1)
x x
x x
x x





 
 
 

14. Galerkin Method Example
 Step 2.
The “weight functions” are the same as
the basis functions
Step 3.
Substitute the trial function y(x) into
i
0
1
, [ ( )] ( ){ [ ( ) ( )]} 0
nb
j j j ja
j
w R D u dx x D x c x dx   

     

15. Galerkin Method Example
 Step 4.
i=1,2,3; we have three equations with
three unknown coefficients1 2 3, ,c c c
31 2
31 2
31 2
43 51
0
15 10 84 315
615 111
0
70 84 630 13860
734 611
0
315 315 13860 60060
cc c
cc c
cc c
    
   
    

16. Galerkin Method Example
 Step 5.
Solve this linear equation set, get:
Obtain the approximation solution
1
2
3
1370
0.18521
7397
50688
0.185203
273689
132
0.00626989
21053
c
c
c
   
 
   
3
1
( ) ( )i i
i
y x c x

 

17. Galerkin Method Example
GalerkinGalerkin solutionsolution Analytic solutionAnalytic solution

18. References
 1. O. C. Zienkiewicz, R. L. Taylor, Finite
Element Method, Vol 1, The Basis, 2000
 2. Galerkin method, Wikipedia:
http://en.wikipedia.org/wiki/Galerkin_method#cite_note-BrennerScott-1