Even among finite difference methods there are different stability criteria. The one you quote is an approximation to that for the use of Euler's method for timestepping when using second order central central differences in space. Backward Euler and Crank Nicholson are unconditionally stable. One may use a fourth order Runge-Kutta scheme for the timestepping and this will be different again.

How can i find the stability condition for 1-D unsteady heat conduction in finite element representation?

As previously addressed, a stability constraint depends on the type of discretization and is not unique for any FD (or FV, FEM, et) discretization.

You can do a stability analysis by looking to the matrix of the discrete scheme and using the Gershgorin localization method for the eigenvalues or performing the von Neumann analysis.

Concerning the FEM, you have to consider not only the type of spatial discretization but the combination with the discrete time integration.

Thanks a lot

I find out what you say but i still have problem with FEM stability analysis

Where can i find a simple example for stability analysis in FEM?

Have a look to the FEM chapter in the book of Chung, Computational Fluid Dynamics.

In a semi-discrete approach you introduce the FEM for the spatial term and then you have an ODE equation so that you can use the classical numerical stability analysis

Dear Steffen, regarding stability conditions for FD and FEM see also

A.Quarteroni, Numerical Models for differential problems

Gianluca

If you use a standard finite element method with linear elements for the space part you end up with a sytem of the form

M du/dt = Su (+ f(t))

with M and S tridiagonal matrices and the f term onluy if you have outside heat sources.

It is almost a no brainer to use an implicit method (Euler implicit or Crank Nicholson) for a system like that, because it is so easy to solve tridiagonal systems, which gives you absolute stability.

If, however, you insist on using an explicit method you can safely use

the same criterion you use for finite differences but take the minimum delta_x. Better is to find the eigenvalue of minimum absolute value of the system S-e M, say e_0 and use the criterion delta_t < e_0

Both solutions will reach to the same equation beyond this point the stability of the solutions are the same.