weak form for advection-diffusion equation

[Jaekwang Kim]

[Jaekwang Kim]

Adding up to this, I was trying as...

scratch_data.fe_values.shape_grad_grad (j,q_point)

but keep receiving errors 

/Users/kimjaekwang/repos/advection_diffusion/step9/step-9.cc:487:53: error: no member named 'shape_grad_grad' in

      'dealii::FEValues<2, 2>'

                           - scratch_data.fe_values.shape_grad_grad (j,q_point)

point)

[Bruno Turcksin]

Jaekwang,

in step-9 the test function is a little bit strange because if you use continuous finite element to discretize the advection equation, the scheme is unstable. So you basically add a diffusion term to the equation to stabilize the scheme. However, your equation already has a diffusion term so you may not need a stabilization term.

Best,

Bruno

On Wednesday, June 14, 2017 at 3:33:18 PM UTC-4, Jaekwang Kim wrote:

[Jaekwang Kim]

Thanks for the response! 

I've read some articles that the governing equation (advection + diffusion) is unstable when it has large Peclet number.

(Even though it has diffusion term.)

What I am concerning is that it is how to imply SUPG stabilization method in deal.ii., because my future problem will include advection-diffusion term....

In SUPG, people are considering residual term ( b dot nabla - c nabla^2 u -f )

.. and I thought this is very closely related to step-9 formation. 

But I am not still sure to assemble residual term in my weak form... 

2017년 6월 14일 수요일 오후 3시 40분 50초 UTC-5, Bruno Turcksin 님의 말:

[Bruno Turcksin]

Jaekwang,

shape_grad_grad doesn't exist but shape_hessian does http://dealii.org/8.5.0/doxygen/deal.II/classFEValuesBase.html#a294fe8cefce05f98a6135582cacf91c5 I think that is what you want.

Best,

Bruno

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[Wolfgang Bangerth]

On 06/14/2017 04:34 PM, Jaekwang Kim wrote:
>
> I've read some articles that the governing equation (advection +
> diffusion) is unstable when it has large Peclet number.
> (Even though it has diffusion term.)
>
> What I am concerning is that it is how to imply SUPG stabilization
> method in deal.ii., because my future problem will include
> advection-diffusion term....
>

> In SUPG, people are considering residual term*( b dot nabla - c nabla^2
> u -f )*

>
> .. and I thought this is very closely related to step-9 formation.
>
> But I am not still sure to assemble residual term in my weak form...

Jaekwang,
it is correct that you do need the stabilization (SUPG or whatever else
you want to use) for the advection as long as advection dominates
diffusion -- i.e., as long as the *local Peclet* number is large.

I am certain that there are many examples in the literature where you
can see this, but my intuition would be that you write down the usual
SUPG scheme for the advection terms, and then you just add the diffusion
term (without any modification) to the resulting bilinear form. I.e.,
the SUPG stabilization does not see and does not know about the
diffusion term.

Best
W.
--
------------------------------------------------------------------------
Wolfgang Bangerth email: bang...@colostate.edu
www: http://www.math.colostate.edu/~bangerth/