advection equation weak form in tutorial 9

[Jaekwang Kim]

Hi, all 

I am little bit confused by weak form for advection problem written in step-9 tutorial 

in the its tutorial, in the middle of the line, it says that... 

I am little bit unsure the red-marked form, 

From the whole boundary integral, we put inflow case, to the right hand side, and remaining part on left-hand-side is cases other than inflow condition 

so I thought, this should be.. 

 

the left hand side term is boundary that is not inflow case... 

I saw similar expression, in Dr. Bangerth video lecture 31, on hyperbolic equation, here his weak form also marked 'not inflow boundary' remaining in left hand side and inflow case, with boundary value description went to RHS

..... I am not sure i am just confusing some of indication... .. 

thanks !

This is a point we made in the introduction of step-3. There, we argued that to avoid this very kind of problem, one should get in the habit of always multiplying with test functions from the left instead of from the right to obtain the correct matrix right away. In order to obtain the form of the linear system that we need, it is therefore best to rewrite the weak formulation to 

(vh+δβ⋅∇vh,β⋅∇uh)Ω−(β⋅nvh,uh)∂Ω−=(vh+δβ⋅∇vh,f)Ω−(β⋅nvh,g)∂Ω−

and then to obtain

[Wolfgang Bangerth]

Jaekwang,

> I am little bit confused by weak form for advection problem written in step-9
> tutorial
>
> in the its tutorial, in the middle of the line, it says that...
>
> I am little bit unsure the red-marked form,

I think you forgot the formula.

> From the whole boundary integral, we put inflow case, to the right hand side,
> and remaining part on left-hand-side is cases other than inflow condition
>
> so I thought, this should be..
>

> <https://lh3.googleusercontent.com/-_-hS916j4W8/WZICJB_jYWI/AAAAAAAABEU/scEwE0R6BlYkOd4_vFYaAm-8rLuEOdqoACLcBGAs/s1600/Screen%2BShot%2B2017-08-14%2Bat%2B2.59.36%2BPM.png>

>
>
> the left hand side term is boundary that is not inflow case...

I'm not entirely sure I understand your question, but you arrive at the weak
formulation by
* taking the original PDE, multiplying it with the modified test functions,
and integrating by parts
* on both sides of the equations, add the term
(beta.n v_h, u_h)_{inflow boundary}
* on the right hand side, replace u_h by g_h on the inflow boundary part

The key step that may not have been obvious is that in the second step, we add
the same thing to both sides of the equation. The equation therefore must
remain the same.

Best
W.

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

[Jaekwang Kim]

Thanks, now I clearly understand what's going on!

Jaekwang