# Stability analysis for explicit time discretization in the Finite Element Method

I have been looking for stability analysis of general reaction-diffusion problems, of the form

$\frac{\partial u}{\partial t}=\nabla\cdot D\nabla u-k\,u$ ,

to be solved using the standard Finite Element in 2D with an explicit Euler time stepping. I haven't found any estimate or proof for the stability criteria for $\Delta t$ for this case, most of the literature seems to point to the solution of the Heat equation by Finite Differences in 1D.

I would like to know if someone can point me how to find the stability criterion in this case, especially for quadrilateral elements.

• What's your time stepping scheme? – Wolfgang Bangerth Jun 18 '15 at 2:05
• The standard explicit Euler time stepping. – Robert Sep 20 '15 at 16:18
• The reaction term adds no complications since its eigenvalues are all proportional to $k$. What matters is the largest eigenvalue, which increases with the mesh resolution, and this largest eigenvalue is only determined by the diffusion term. – Wolfgang Bangerth Sep 22 '15 at 21:59