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Computational Engineering
Galerkin Method
Yijian Zhan
Ning Ma
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
Galerkin Method
In practical cases we often apply
approximation.
One of the approximation methods:
Galerkin Method, invented by Russian
mathematician Boris Grigoryevich Galerkin.
Galerkin MethodRelated knowledge
Inner product of functions
Basis of a vector space of functions
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
af g f x g x dx
, 0f g
,w f
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
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{ ( )}ni ix
( ) ( ( )) ( ) 0D U L U x f x
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
Galerkin MethodWeighted 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)
, ( ) , ( ) ( ){ [ ( )]} 0b
aw R x w D u w x D u x dx
Galerkin MethodWeighted 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
, [ ( )] ( ){ [ ( ) ( )]} 0nb
j j j ja
j
w R D u dx x D x c x dx
jc
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
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
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
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
, [ ( )] ( ){ [ ( ) ( )]} 0nb
j j j ja
j
w R D u dx x D x c x dx
Galerkin Method Example
Step 4.
i=1,2,3; we have three equations with
three unknown coefficients 1 2 3, ,c c c
31 2
31 2
31 2
43 510
15 10 84 315
615 1110
70 84 630 13860
734 6110
315 315 13860 60060
cc c
cc c
cc c
Galerkin Method Example
Step 5.
Solve this linear equation set, get:
Obtain the approximation solution
1
2
3
13700.18521
7397
506880.185203
273689
1320.00626989
21053
c
c
c
3
1
( ) ( )i i
i
y x c x
Galerkin Method Example
GalerkinGalerkin solutionsolution Analytic solutionAnalytic solution
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