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[Getfem-users] Fokker-Plank equation

From: Magnus Paulsson
Subject: [Getfem-users] Fokker-Plank equation
Date: Thu, 19 May 2011 14:47:49 +0200

Dear GetFEM++ users,

I would like to play around with transport equations and while writing the 
questions below I realised that the stiffness matrix will not be 
symmetrical. So before we start: can GetFEM calculate the non-symmetric 
stiffness matrix? There is no need to solve the linear system. From the 
mailing list it seems that the answer is no.

NO: can you suggest any other FEM package that can handle 
these kind of equations in more than 3 dimensions?

YES: Please read on.

I would like to play around a little bit with the Fokker-Planck (FP) 
equation: (a PDE that has many names so you may recognise is as (at 
least similar to) Klein-Kramers, Liouville, Boltzmann-transport 
equation etc ...). However, I have limited experience with FEM and 
thus problems in deciding whether or not FEM is the way to go and to 
decipher the user manual.

The FP can be used to describe Brownian motion of particles.

u(x,p) = probability to find particle at x with momentum p.

2-D PED (one particle with position and momentum):
du/dt = dV(x)/dx du/dp - p/m du/dx + beta d/dp[u p + m kT du/dp]

V(x) : the potential energy of the partice (dV/dx = - force), scalar field 
       e.g., x^2 for a harmonic potential
m    : mass, scalar
beta : friction, scalar
kT   : temperature, scalar

The main question:
0: Can the GetFEM++ handle the first order derivatives? Do they not give a 
non-symmetric stiffness matrix?
1: How do I enter the first order derivative terms, i.e., A.grad(u) terms?
Where A is a vector or vector field. (Preferentially in the python 

Thanks - Magnus

Magnus Paulsson
Assistant Professor
School of Computer Science, Physics and Mathematics 
Linnaeus University
Phone: +46-480-446308
Mobile: +46-70-6942987 

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