advection-diffusion demo

Question

Hello, I'm trying to understand the supg approach in the advection-diffusion demo: https://github.com/FEniCS/dolfin/blob/master/demo/undocumented/advection-diffusion/python/demo_advection-diffusion.py

My question is the following: the term div(grad(u_mid)) in the "residuum" seems to be always zero (the functions are piecewise-linear) (line 72 and 79). I don't see that this "residuum" term should converge to zero with fine mesh and small time-step. Everything made perfect sense in a space of functions that are C1-continuous, but is there a reason to assume that the current implementation (using FunctionsSpace(CG, 1)) approximates a real solution in some way? Moreover, why to write "div(grad(u_mid))" when it is zero?

Thanks a lot, Peter

Answer 1

Well, you're right that the term div(grad(u_mid)) is redundant for CG1 elements. But it is included to keep r looking as the residual of the strong advection-diffusion equation. You can also increase order of an element and r will still be valid.

Added stabilization term is consistent, i.e. it takes zero value for exact solution. You can derive it by integration by parts on every cell separately. (Note that you are not able to do this integration by parts globally because you lack regularity. But stabilization term is just cell-wise integral - it does not take into account potential Dirac-$\delta$ on facets.)

Further reading:

• T. E. TEZDUYAR: Stabilized Finite Element Formulations for Incompressible Flow Computations. ADVANCES IN APPLIED MECHANICS. VOLUME 28 (1992) 1-44.

• Tayfun Tezduyar and Sunil Sathe: STABILIZATION PARAMETERS IN SUPG AND PSPG
  FORMULATIONS. Journal of Computational and Applied Mechanics, Vol. 4., No. 1., (2003), pp. 71-88.