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I will approach this from the perspective of using Galerkin methods for
approximating PDEs using fnite element techniques.

The Galerkin is an alternative formulation to the variational formulation
typically used when a variational does not exist.  If it is possible to integrate
the Galerkin form by parts one typically ends up with a variational formulation.

This statement has the following consequences:
 
     - a pure Galerkin formulatiuon typically results in nonsymmetric equations

     - a pure Galerkin forumlation typically requires trial functions of twiceUnread news in news.config                                 1 article  +  412 oldUnread news in news.config                                 1 article  +  412 old
            the order of an equivalent Variational formulation

There is also the issue that a Galerkin forumation, since it need not be
positive definite, can result in saddle solutions rather than the minima typical
of variational forms.  This has the effect in certain nonlinear PDEs for
introducing instability, although I am not aware of solid research to support
this statement for the general case,  It would  probably be an interesting
problem.  (Careful though, when evaluating convegence properties one should look
at the convergence of primary variables only, not secondary variables introduced
to enforce boundary conditions - e.g. one should not look at the convergence of
problems with Lagrange multiplier constraints by by including the behavior of the
Lagrange multipliers in the convergence study).

Hope this is useful!
-- 

Regards.
 
Don Rolph  a722756@pan.mc.ti.com WD3 MS10-13 (508)-699-1263

In article <yg!31aq@rpi.edu> sassoj@aix.rpi.edu (John J. Sasso Jr.) writes:
>
>    The Galerkin method, which is used to approximate solutions to differential
>   equations, appears to be a pretty interesting technique.  However, I am 
>   not too knowledgable about it, so I'd like to know what is its advantages
>   and disadvantages.  Also, how does it fair against nonlinear ODEs and PDEs,
>   which tend to have chaotic solutions.  If anyone has any good introductory
>   references to the Galerkin method, I would like to know about them.
>
>                          -- John

Galerkin method is a particular type of weighted residual technique in
which the shape (trial) functions themselves are used as weighting
functions. There are several different types of weighted residual methods
such as point collocation, subdomain collocation, least squares, and
Galerkin. These are all powerful techniques, and are very simple to
implement in finite element analysis because you don't have to find a
variational formulation for nonstructural problems. Galerkin method is the
most popular weighted residual method in finite element analysis, and it
can be proved that if the problem has a classical variational form,
Galerkin method will give the exact same results as a variational technique
such as Rayleigh-Ritz.

Look in "The Finite Element Method" by Zienkiewicz for more details about
this method.
-- 
Pardeep K. Bhatti                         pkbhatti@kopth004.delcoelect.com
Delco Electronics Corporation
GM Hughes Electronics
***  The opinions expressed are mine, not necessarily of my employer.  ***

In a previous article, sassoj@aix.rpi.edu (John J. Sasso Jr.) says:

>    The Galerkin method, which is used to approximate solutions to differential
>   equations, appears to be a pretty interesting technique.  However, I am 
>   not too knowledgable about it, so I'd like to know what is its advantages
>   and disadvantages.  Also, how does it fair against nonlinear ODEs and PDEs,
>   which tend to have chaotic solutions.  If anyone has any good introductory
>   references to the Galerkin method, I would like to know about them.
>
>                          -- John
>
   My introduction to the Galerkin method as appled to a solution of
electromagnetic wave propagation in a plasma was the article
TEXTBOOK FINITE ELEMENT METHODS APPLIED TO LINEAR WAVE PROPAGATION
PROBLEMS INVOLVING CONVERSION AND ABSORPTION by Appert et al in
Computer Physics Comm. vol.40(1986)pp 73--93.  

Section 2 describes the physics and diff eq. sect.3 deals with the application
of the Galerkin method to the problem.  It took several readings, but was
worth it.
-- 
Mirko Vukovic 
University of Wisconsin -- Madison
VUKOVIC@UWMFE.NEEP.WISC.EDU 

Galerkin methods with finite elements are very widely used for parabolic systems like
Navire-Stokes. The conventional Galerkin Finite Elements (GFEM) are as estable for these
systems.  However, for predominantly hyperbolic systems (Euler Equations, Vlasov Equations),
GFEM loses its stability.  As a result, some form of stabilization is necessary like
a streamline upwinding has to be added.  There are also other ways to stabilize the GFEM
for hyperbolic systemsby using the so-called Petrov-Glaerkin techniques in which you
have the rial functions different from the test functions.  SOme error estimates are
available for different methods (Claes Johnson (1986), T.J. Hughes (1982)) and so on.

	Now for non-linear systems; It is well known that Navier-Stokes equations can
be primary source of chaos under appropriate parameter conditions (the Lorenz equations).
It is possible to use GFEM for such systems as well. However, it should be noted that
chaotic behavior is a characteristic of the "system of equations that one is solving" and
will manifest independent of the numerical technique.  Appropriate manipulation of the
same technique (GFEM for instanace) should not be a problem to study this behavior.

	It has also been observed that systems of equations can change their behavior
inside the domain.  FOr example parabolic systems can be come hyperbolic (boundary layer
flow, non-newtonian polymeric fluids).  In cases like these, GFEM (assuming that stabilizing
properties are already incoporated) can be used.  
	Hope that the above lines answer at least some of your questions.

S.Shankar
Tech CAD
INTEL Corporation

In article <yg!31aq@rpi.edu>, sassoj@aix.rpi.edu (John J. Sasso Jr.) writes:
|>     The Galerkin method, which is used to approximate solutions to differential
|>    equations, appears to be a pretty interesting technique.  However, I am 
|>    not too knowledgable about it, so I'd like to know what is its advantages
|>    and disadvantages.  Also, how does it fair against nonlinear ODEs and PDEs,
|>    which tend to have chaotic solutions.  If anyone has any good introductory
|>    references to the Galerkin method, I would like to know about them.

Take a look at "Variational Methods in Mathematics, Science and Engineering"
by Karel Rektorys, for further explanation and comparison with other methods.

Larry Edwards

Some general references for Galerkin techniques

	Computational Galerkin Methods by C.A.J. Fletcher

	Numerical Methods in Partial Differential Equations by L.Lapidus and Pinder

	Finite Element Method  G. Strang and Fix

	Non-linear Methods ..... B. Finlayson

There are other authors like Zeinkinowicz (sp ?), Hughes, and so on.  The book by
Strang  and the one by Finlayson are well worth looking into, for good background and a
wider perspective.

S.Shankar
Tech CAD
INTEL