Spectral Methods using dealii

[陈敏]

Dear all

 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? Or Are there some misunderstanding of Spectral Methods?

Best

Chen

[Wolfgang Bangerth]

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

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

[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]:

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.

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 <bang...@colostate.edu> 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

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

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[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 <hkch...@gmail.com> 于2024年6月23日周日 18:53写道：

[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.
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