Re: [Getfem-users] Integro-differential equations

[Yves Renard]

Dear Ivan,

Yes, if you can expand K(x,x') in \sum (a_k b_l \phi_k(x') \phi_l(x)), you can 
compute two separate matrices and obten what you want. But this is a case where 
you can approximate  K(x,x') with a finite element method, which does not 
correspond to most of interesting cases where the kernel has singularities ... 
you are right.

> To do it I need some data from FEM. Exactly, I need base functions and 
> coordinates of
> area it's defined on. Can getFEM provide this data?

You can obtain the expression of shape function on the reference element and 
the expression of the transformation to the real element (in case of affine 
transformations). You can also interpolate the shape functions on any points.

Yves. 


----- Original Message -----
From: "Ivan Melikhov" <address@hidden>
To: "Yves Renard" <address@hidden>
Cc: address@hidden
Sent: Friday, January 18, 2013 7:25:40 PM
Subject: Re: [Getfem-users] Integro-differential equations

Yves, 


Thank you for your response. 
The integral in my previous message is correct. But forget it. Let the 1D 
equation involves term \int(K(x,x')u(x'))dx' where K is known kernel and u is 
unknown function we are solving for. So to solve it with FEM, one needs to 
assemble matrix \int\int(K(x,x')\phi_i(x')phi_j(x))dx'dx. Do you mean that I 
can expand K(x,x') in \sum (a_k b_l \phi_k(x') \phi_l(x)), compute two 
separable integral and multiply their values? 


Another problem is that my kernel has singularity, so it isn't interpolated 
well by polynomial base functions. The best way I see is to compute the whole 
matrix not in getFEM but in matlab or mathematica. To do it I need some data 
from FEM. Exactly, I need base functions and coordinates of area it's defined 
on. Can getFEM provide this data? 


Thanks, 
Ivan 



2013/1/18 Yves Renard < address@hidden > 





Dear Ivan, 

Unfortunately, the assembly procedure of Getfem is not designed to compute such 
integro-differential term. 
May be if you have specific expression for the kernel (if it is simple or can 
be expressed on a FEM) it should be possible to adapt something. 

Yves. 


Le 18/01/2013 12:24, Ivan Melikhov a écrit : 



Hello! 


I need to solve an integro-differential eigenvalue problem and I have trouble 
with integral term. Generally, I need to assemble a 4D matrix with elements 
\int\int(\phi_i(r')\phi_j(r')V(r,r')\phi_m(r)\phi_l(r))dr'dr I suppose it is 
\int\int(\phi_i(r)\phi_j(r')V(r,r')\phi_m(r)\phi_l(r))dr'dr (\phi_i(r) instead 
of \phi_i(r')) ? 






where V(r,r') is known function, \phi_i is the ith base function, dr is dxdy, 
dr' is dx'dy'. The question is how can I compute an inner integral inside comp 
command in generic_assembly::set? 


Thank you for answers, 
Ivan 

_______________________________________________
Getfem-users mailing list address@hidden 
https://mail.gna.org/listinfo/getfem-users 

-- 

  Yves Renard ( address@hidden )       tel : (33) 04.72.43.87.08
  Pole de Mathematiques, INSA-Lyon             fax : (33) 04.72.43.85.29
  20, rue Albert Einstein
  69621 Villeurbanne Cedex, FRANCE http://math.univ-lyon1.fr/~renard ---------