[Getdp] WG: Formulation

[Kubicek Bernhard]

Hello!

Maybe this mail I once wrote helps you. http://www.geuz.org/pipermail/getdp/2006/000864.html

The main idea with weak form is this (at least as I understand it): You have a differential equation of second order. You want to solve it with picewise linear functions. This is a problem, because the  second derivative is zero within the elements (e.g. Volumetric Cells), or undefined at the one-less-dimensional connection elements (e.g. Surfaces of cells). The solution is the weak form, also sometimes called "variational formulation" e.g. see: http://en.wikipedia.org/wiki/FEM#The_variational_form_of_P2

(basically thats just a 3D partial integration)

Another source is the often cited Bossavit (see Wiki->Theoretical References, the pdf is online somewhere), which however is written for mathematicians.

Understanding the weak form _is_ somewhat necessary for creating new formulations in GetDP. But no worries, if you digg in the archives, you will find that nearly everyone new to GetDP first bumps against this step. 

nice greetings,
Bernhard

PS: Sorry for maybe sounding sarcastic and bad spelling today, but I spend the last 11 hours reprogramming FLUENT UDFs (c-sourecode plugins) for them to work in parallel on a cluster. And believe me, _that_ is quite frustrating; e.g. you cannot use // for commenting, only /* . And if there is an error in your source, you have to waste 2 minutes setting everything up again, so it can basically crash again.

-----Ursprüngliche Nachricht-----
Von: getdp-bounces at geuz.org [mailto:getdp-bounces at geuz.org] Im Auftrag von Tammo Heeren
Gesendet: Mittwoch, 10. Januar 2007 17:50
An: getdp at geuz.org
Betreff: Re: [Getdp] Formulation


So,
- if we have scalars ( I assume given by 'Type Form0' in the  
FunctionSpace definition, '{d scalar}' would results in 'grad scalar'.
- 'Dof' indicates the unkown parameter (space)

Where is the 'div' operation indicated?
What does the second {d v} mean?

Tammo


-----------------------

The expression
Galerkin { [ epsr[]*Dof{d v} , {d v} ] ; In Vol; Jacobian Vol ;   
Integration Int ; } Is a way to describe the Galerkin formulation of  
the problem. It is the variationnal formulation of the problem using  
Galerkin's assumption. You can see for example section Variationnal  
Formulation there:
http://www.geuz.org/getdp/wiki/WaveEquationPml     (username:getp   
password:wiki)

The expression Dof{athing} means that athing is the unknown parameter  
(Dof: Degree Of Freedom)

The sign d is the Exterior derivative: applied to a p-form, gives a  
(p+1)-form. This means that applied to a scalar, d is the gradient.

> I need some help in understanding formulations. How does the following
> line: "Galerkin { [ epsr[]*Dof{d v} , {d v} ] ; In Vol; Jacobian Vol
> ; Integration Int ; }" translate into "div epsr[] grad v=0". Is the 
> grad operation implicit in "Dof{d v}"? Where is the div operation 
> specified? These are probably very simply questions, maybe someone can 
> enlighten me.
>
> Tammo

_______________________________________________
getdp mailing list
getdp at geuz.org
http://www.geuz.org/mailman/listinfo/getdp