Re: [Libmesh-users] [RB] Min-theta approach

[David K. <dav...@ak...>]

On Tue, Apr 3, 2018 at 4:58 AM, <ss...@pu...> wrote:

> Thank you for your reply, David.
>
>
>
> I understood the answer as follows: ThetaA1 and ThetaA2 can be neglected
> in calculating the coercivity lower bound due to the divergence-free
> convection field.
>
> However, I am not familiar with the field of thermal fluid engineering, so
> I do not know in detail why it can be ignored.
>
>
>
> Could you tell me more about this problem with formulations of the lower
> bound and the divergence-free convection field?
>
> Or can you tell me about references related to this problem?
>
>
>
> I always appreciate your help.
>


This follows from the definition of the coercivity constant. Set trial and
test function to be the same in the convection-diffusion bilinear form, and
integrate the convection term by parts and you end up with a term that
includes the divergence of the velocity field. If the velocity field is
divergence-free, then that term vanishes.

David


*From:* David Knezevic <dav...@ak...>
*Sent:* Monday, April 2, 2018 8:34 PM
*To:* 강신성 <ss...@pu...>
*Cc:* libmesh-users <lib...@li...>
*Subject:* Re: [Libmesh-users] [RB] Min-theta approach



On Sun, Apr 1, 2018 at 10:26 PM, <ss...@pu...> wrote:

Hello, all.



I try to derive an RB error bound using the min-theta approach.

First of all, I saw the RB example 1 because this example shows the value of
coercivity constant lower bound, not dummy value.

However, this example does not satisfy requirements of the min-theta
approach.



If so, how was the lower bound value of the example 1 obtained?

I do not know why the lower bound value is 0.05 in the RB example 1.





If I recall correctly, the ThetaA1 and ThetaA2 are irrelevant to the
coercivity constant because they give a divergence-free convection field
(it's clearly divergence free since the field is constant everywhere, given
by the parameters x_vel and y_vel). As a result we can set the coercivty
lower bound to be the value returned by ThetaA0, which is 0.05.



Of course, this is a simple case, and in general one must use SCM to get
the coercivity lower bound. I would say that in practice a lot of people
are satisfied with skipping the SCM and just setting a dummy value (e.g. 1)
for the coercivity lower bound. This means that the error bound isn't
rigorous anymore, but it's still useful as an error indicator.



David