Having trouble with FEM functional

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3 total messages Started by chakravaa Sun, 27 Dec 1992 03:19

#884

Author: chakravaa
Date: Sun, 27 Dec 1992 03:19

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Good day,
   I'm interested in solving a Diffusion equation problem using FEM.  However,
I'm having difficulty putting the equation into a functional form that can be
manipulated.  The equation is a 1-D equation with the following boundary
conditions:
       u(x,0) = f(x) = sin x
and    u(0,t) = u(L,t) = 0  as bc's.
the actual equation as such comes to a solution of
   u(x,t) = sin (x) (e**(-k**2 t )
when solved analytically.
When attempting to use the Euler-Lagrange eqn. to find the functional form,
I was having extreme difficulties in determining my course of action,
particularly in solving for adjoint operators in the equation.
Any help would be appreciated.
 
Ut = k**2 Uxx is the Diffusion equation.

Thanx.
A.C.

Re: Having trouble with FEM functional

#911

Author: Pierre Asselin
Date: Mon, 28 Dec 1992 20:47

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In <1992Dec26.221924.1@woods.ulowell.edu> chakravaa@woods.ulowell.edu writes:

>Good day,
>   I'm interested in solving a Diffusion equation problem using FEM.
>[...]
>I was having extreme difficulties in determining my course of action,
>particularly in solving for adjoint operators in the equation.

The equation is not self-adjoint.  One way to proceed it is to
discretize only the space part of the problem with finite elements.
This leaves you with a system of coupled ordinary differential
equations in time, wich you solve with finite differences.  If you want
unconditional stability for all time step sizes, you will need an
implicit scheme.  See Strang and Fix, chapter 7.
-- 

--Pierre Asselin, Magnetoresistive Head Engineering, Applied Magnetics.
  pa@appmag.com        the usual disclaimers apply.

Re: Having trouble with FEM functional

#918

Author: Sadasivan Shankar
Date: Tue, 29 Dec 1992 18:24

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The FEM can be very efficiently used for parabolic equations.  In fact, exhaustive analysis
have been done for parabolic systems. As suggested, the method of lines (space-part by
FEM and time-part by finite differences) is the most common way of doing it.  However, there
are other ways to approach the problem.  For example, the group led by Claes Johnson at
Chalmers University, Sweden have used the so-called space-time formulation for diffusion
equations and have shown that the scheme is of higher-order accuracy (than MOL) and stable.
Refer to papers in Computer Methods in Applied Mechanics and Engineering . In these techniques,
the time-dimension is also approxmated the same way as space-dimension.

S.Shankar
Tech CAD
INTEL

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