# 2d advection-diffusion: cell Péclet number and numerical stability

I am studying the numerical resolution of 2d advection-diffusion problems with finite element methods.

$$\frac{\partial u}{\partial t} + \beta\cdot\nabla u = \nabla\cdot(\nabla u) \, .$$

It is said in this article on the COMSOL blog that it has been mathematically proven that the condition on the cell Péclet number

$$\mathrm{Pe}:= \frac{\Vert \beta\Vert h}{2c} > 1\, ,$$

leads to numerical instabilities.

Could you please point me to any reference rigorously justifying such a statement?

• That depends very much on (i) how you define "instabilities" and (ii) what time and space discretization you use. I would recommend the book by Elman, Sylvester and Wathen for these matters. Sep 7 '21 at 19:21
• Have a look in the notes of my lecturer: aurora.asc.tuwien.ac.at/~plederer/lec_notes/fluid_notes.pdf . In the chapter 3.2 Approximation of scalar convection-diffusion equations, you find at page 88 some notes on the Péclet number. You essentially get this dependence by applying Cea’s lemma. The assumption here is that you have a conforming finite element space.
– Pepe
Sep 7 '21 at 21:33
• I found those notes very interesting. Thanks very much. Sep 17 '21 at 13:16