sci.math.numanalysis
SUNA41, 1D introduction to FEM
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A good, readable book to start with is: Norrie D.H. & de Vries G.;
"An Introduction to Finite Element Analysis"; Acad. Press 1978.
However, only the first few chapters of it are truly usefull, IMHO.
The following example could easily be worked out in a FEM class.
Consider the equation: dQ/dx = 0 where Q = C(x).dT/dx
It describes for example heat conduction in a rod:
x = coordinate , Q = heat flow , T = temperature , C = conductivity .
Some understanding of variational principles is required in the first place,
because the standard approach to Finite Elements is always via such principles.
For the problem at hand:
/
 C(x).(dT/dx)^2.dx = minimum ; integral is taken over the whole 1D grid
/ ***********
1 2 3 4 5 6 7 8 ...
Next split it in pieces over linear elements, with shape functions (h, 1h),
where x = (1h).x1 + h.x2 . 1**2
A little algebra gives: h = (xx1)/(x2x1)
The same for temperatures: T = (1h).T1 + h.T2 and C(h) = (1h).C1 + h.C2
This simplifies calculation of the integrals:
/1
 C(h).(dT/dh.dh/dx)^2.(x2x1).dh = (C1+C2)/2 . {(T2T1)/(x2x1)}^2.(x2x1)
0/ _______constant________
Here we neglected the boundary conditions. The _SUM over_ all these algebraic
expressions for all elements must reach a minimum, as a function of T1 and T2.
Differentiate (partial) to the latter, everywhere on the grid:
d/dT1 > + (C1+C2)/(x2x1) . T1  (C1+C2)/(x2x1) . T2
d/dT2 > + (C1+C2)/(x2x1) . T2  (C1+C2)/(x2x1) . T1
Rearranging:
 +1 1  T1 
SUM (C1+C2)/(x2x1) .    = 0 (: matrix!)
elements  1 +1  T2 
1 2 3 : global
Assemble TWO of these for neighbouring elements; *A*B*
don't forget to change local into global numbers: 1 2+1 2 : local
 (C1+C2)/(x2x1).T1 + {(C1+C2)/(x2x1)+(C2+C3)/(x3x2)}.T2  (C2+C3)/(x3x2).T3
A A B B
For a point off the boundary, this expression must be zero: an "F.D." scheme.
If specialized for (x2x1) = (x3x2) = 1 , then: ^^^^
 (C1 + C2).T1 + (C1 + 2.C2 + C3).T2  (C2 + C3).T3 = 0
Which EXACTLY matches the (equivalent) Finite Volume scheme for this problem.
Which is NO coincidence at all ...

* Han de Bruijn; Applications&Graphics  "A little bit of Physics * No
* TUD Computing Centre; P.O. Box 354  would be NO idleness in * Oil
* 2600 AJ Delft; The Netherlands.  Mathematics" (HdB). * for
* Email: Han.deBruijn@RC.TUDelft.NL  Fax: +31 15 78 37 87 * Blood