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Re: [Getfem-users] Integro-differential equations


From: Ivan Melikhov
Subject: Re: [Getfem-users] Integro-differential equations
Date: Fri, 18 Jan 2013 22:25:40 +0400

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


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