Re: [deal.II] Spectral Methods using dealii

[陈敏]

Dear Wolfgang Bangerth

I learn spectral methods from the book "Hesthaven, Nodal Discontinuous
Galerkin Methods, 2008". This book focuses on the Nodal Discontinuous
Galerkin Methods, and use Legendre polynomials to expanse the solution of a
nodal element on a polynomial space like the followed equations[Page 53 of
the book]:
[image: image.png]
ψn is the Legendre polynomial basis on polynomial space, and *lik is
the *lagrangian
polynomial basis on physical space. There will be a transform between the
two spaces by projection.

Anyway, if I want to implement a spectral method in dealii, I just need to
use Legendre polynomial basis and do quadrature on Legendre-Gauss-Lobatto
(LGL) quadrature points?

Thanks!

Best
Chen

Wolfgang Bangerth  于2024年6月19日周三 07:08写道：

> On 6/15/24 09:51, 陈敏 wrote:
> >
> >   I have a question of Spectral Methods using dealii. As far as I know,
> we can
> > you the Legendre function and some other Orthogonal polynomials as basis
> > function in dealii. But we need the projection from polynomial space to
> > physical space, how to do the projection in dealii?
>
> I am not familiar with the term "projection from polynomial space to
> physical
> space". Do you mean the transformation from reference cell to a particular
> cell of the mesh? If not, can you elaborate?
>
> Best
>   Wolfgang
>
> –
> --
> Wolfgang Bangerth  email: bange...@colostate.edu
> www: http://www.math.colostate.edu/~bangerth/
>
>
> –
> The deal.II project is located at http://www.dealii.org/
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> .
>

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Re: [deal.II] Spectral Methods using dealii

[陈敏]

Dear Wolfgang Bangerth

I found "FE_DGQLegendre" in dealii, and I think it is the spectral methods
but use discontinous Galerkin. I also found "FESeries::Legendre", this
class provide the transformation

between the  polynomial space and physical space. But it lack tutorial of
the two classes.

Best
Chen

陈敏  于2024年6月23日周日 18:53写道：

> Dear Wolfgang Bangerth
>
> I learn spectral methods from the book "Hesthaven, Nodal Discontinuous
> Galerkin Methods, 2008". This book focuses on the Nodal Discontinuous
> Galerkin Methods, and use Legendre polynomials to expanse the solution of a
> nodal element on a polynomial space like the followed equations[Page 53 of
> the book]:
> [image: image.png]
> ψn is the Legendre polynomial basis on polynomial space, and *lik is the 
> *lagrangian
> polynomial basis on physical space. There will be a transform between the
> two spaces by projection.
>
> Anyway, if I want to implement a spectral method in dealii, I just need to
> use Legendre polynomial basis and do quadrature on Legendre-Gauss-Lobatto
> (LGL) quadrature points?
>
> Thanks!
>
> Best
> Chen
>
> Wolfgang Bangerth  于2024年6月19日周三 07:08写道：
>
>> On 6/15/24 09:51, 陈敏 wrote:
>> >
>> >   I have a question of Spectral Methods using dealii. As far as I know,
>> we can
>> > you the Legendre function and some other Orthogonal polynomials as
>> basis
>> > function in dealii. But we need the projection from polynomial space to
>> > physical space, how to do the projection in dealii?
>>
>> I am not familiar with the term "projection from polynomial space to
>> physical
>> space". Do you mean the transformation from reference cell to a
>> particular
>> cell of the mesh? If not, can you elaborate?
>>
>> Best
>>   Wolfgang
>>
>> –
>> --
>> Wolfgang Bangerth  email: bange...@colostate.edu
>> www: http://www.math.colostate.edu/~bangerth/
>>
>>
>> –
>> The deal.II project is located at http://www.dealii.org/
>> For mailing list/forum options, see
>> https://groups.google.com/d/forum/dealii?hl=en
>> —
>> You received this message because you are subscribed to the Google Groups
>> "deal.II User Group" group.
>> To unsubscribe from this group and stop receiving emails from it, send an
>> email to dealii+unsubscr...@googlegroups.com.
>> To view this discussion on the web visit
>> https://groups.google.com/d/msgid/dealii/9cc4ed3b-3c65-4071-b1e8-918de2373470%40colostate.edu
>> .
>>
>

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Re: [deal.II] Finite Volume Method - Matrix/RHS assembly

[Abbas Ballout]

Nick, 

2)
When you use DG0, all of the terms in the weak from cancel to zero except 
for the face term penallty*jump(u)*jump(v).  (du/dx and dv/dy are zero 
inside an element)
That penalty term in the DG community is seen as something that adds 
stability and isn't at the core of the scheme.  

In FV after doing your Gauss integral you end up asking yourself "what is 
the value of the gradient at the face" and set that to be mu*(u2-u1)/h. You 
get to construct one based on the values on an element and it's neighbour
That's a shortcoming of FV in a way because if you want to construct 
anything of higher order you'll have to querry values form the neighbours 
of neighbours of neighbours... In 2d/3d FV and with unstructured grids, 
constructing this isn't easy. 
The second term in the DG weak form: the jump(u) *mu * average(gradv), is 
supposed to be exactly that, but it gets cancelled out when you use DG0. In 
DG grad(v) on a face is calculated based on data of the local cell, in FV 
grad(v) gets calculated based on data from the neighbouring cells. BTW, 
that term is what you get from integrating by parts. 
Back to your question and if you insist on using DG0:
So to match the penalty*jump(u)*jump(v) with what you get from an FV  
scheme (mu*(u2-u1)/h) maybe you might try to set the penalty to be 
something like (mu/h) where h is the distance between the element centers. 
(That's not it tho)
 
1) 
That penalty term is there to enhance stability but I don't know if there 
is more to it. It penalizes the jump across the elements. See if going 
higher with a constant penalty helps with the accuracy.
Or maybe scale it with the element size. 
There are other DG tutorials for the Poisson equation maybe they'll have 
something there for you too. 
(Never liked that penalty term tbh either. All my homies hate the penalty 
parameter. JK). 

Abbas
On Sunday, June 23, 2024 at 1:24:38 AM UTC+2 Wolfgang Bangerth wrote:

>
> Nik:
>
> > (1) trying to solve in 1D:
> > 
> > * I get errors of the following type:
> > *error: ‘class dealii::DoFAccessor<0, 1, 1, false>’ has no member
> > named ‘measure’
> >   411 | re() / cell->face(face_no)->measure();* / I can make it run
> > by removing the calls to these functions and replace the penalty
> > factor which would involve them by a constant number, which according
> > to the output seems to work o.k. but might need further tuning of the
> > penalty constant (here just chosen as a constant of 100)./
>
> Yes, that's because the measure of a face in 1d is hard to define -- what 
> is 
> the measure of a point in a zero-dimensional space?
>
> If you have a good idea for how the 'extent1/2' variables should be 
> defined in 
> 1d, let us know and we can patch the program so that it will also work in 
> 1d.
>
>
> >  (2) trying to solve for polynomial degree 0 (i.e. piecewise constant):
> > 
> > * /When I try to solve the default convergence_rate test case in step-74
> > with degree p = 0, I don't see a decrease in the L2 norm of the solution.
> > I tested different values for the penalty_factor for which the given
> > function would evaluate to 0 because of the p*(p+1) term. I tested just
> > removing the p*(p+1) term and also tested constants like 1000 or 0.001,
> > but in all cases, I got results similar to the table below, whre the L2
> > norm does not decrease. The table below is for the case with penalty term
> > /= 0.5 *(1./cell_extent_left+1./cell_extent_right);
> > 
> > degree = 0
> > | cycle | cells | dofs  | L2| L2...red.rate.log2 | H1| 
> > H1...red.rate.log2 | Energy|
> > | 0 | 16| 16| 3.016e-01 | -  | 4.443e+00 | - 
> >| 6.045e+00 |
> > | 1 | 64| 64| 2.273e-01 | 0.41   | 4.443e+00 | 
> 0.00 
> > | 5.680e+00 |
> > | 2 | 256   | 256   | 2.284e-01 | -0.01  | 4.443e+00 | 
> 0.00 
> > | 5.554e+00 |
> > | 3 | 1024  | 1024  | 2.368e-01 | -0.05  | 4.443e+00 | 
> 0.00 
> > | 5.497e+00 |
> > | 4 | 4096  | 4096  | 2.428e-01 | -0.04  | 4.443e+00 | 
> 0.00 
> > | 5.470e+00 |
> > | 5 | 16384 | 16384 | 2.462e-01 | -0.02  | 4.443e+00 | 
> 0.00 
> > | 5.456e+00 |
> > | 6 | 65536 | 65536 | 2.481e-01 | -0.01  | 4.443e+00 | 
> 0.00 
> > | 5.448e+00 |
> > 
> > Maybe this is well-known and it shouldn't work. However, it would be 
> nice to 
> > be able to solve with degree 0 because that would allow me to mimic a 
> finite 
> > volume type implemenation.
>
> I don't know, and I don't know whether the people who wrote step-74 have 
> thought about the case. One would *hope* that the method also works for 
> the 
> case p=0. A good first step would be to look at how the solution looks if 
> you 
> visualize it -- that is often a good start towards debugging what is going 
> wrong.
>
> Best
> W.
>
> -- 
> ---

Re: [deal.II] Spectral Methods using dealii

[Wolfgang Bangerth]

On 6/23/24 04:53, 陈敏 wrote:


ψ_n is the Legendre polynomial basis on polynomial space, and /l_i ^k is the 
/lagrangian polynomial basis on physical space. There will be a transform 
between the two spaces by projection.

/


I see, yes. This basis transform is facilitated by a simple matrix on each 
cell for which there are functions that can compute it (either in FETools or 
in the finite element itself) if you want to use a *local* projection. If you 
want to transform globally, it requires the inversion of a mass matrix, which 
is also not difficult.



Anyway, if I want to implement a spectral method in dealii, I just need to 
use Legendre polynomial basis and do quadrature on Legendre-Gauss-Lobatto 
(LGL) quadrature points?


Yes, more or less.

Best
 W.
--

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


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