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