[Nek5000-users] Basis functions and smoothness of interpolants

[nek5000-users at lists.mcs.anl.gov]

Thank you for a very good answer.

In the slides by Paul he refers to some MATLAB code examples.
Are these examples available somewhere?

Also: What time discretization is used?


/Johan

________________________________
From: Nek5000-users <nek5000-users-bounces at lists.mcs.anl.gov> on behalf of nek5000-users at lists.mcs.anl.gov <nek5000-users at lists.mcs.anl.gov>
Sent: Thursday, January 4, 2018 5:10:01 PM
To: nek5000-users at lists.mcs.anl.gov
Subject: Re: [Nek5000-users] Basis functions and smoothness of interpolants

Hi,

Indeed, this book is a good resource for learning SEM, and in particular
the implementation in Nek5000. There are also other, perhaps shorter,
expositions of the material. Two papers that I found particularly useful
(there are of course many more) are:

- Fischer. An Overlapping Schwarz Method for Spectral Element Solution
of the Incompressible Navier-Stokes Equations. J. Comput. Phys. 133,
84-101 (1997)
- Fischer et al. Simulation of high-Reynolds number vascular flows.
Comput. Methods Appl. Mech. Engrg. 196 (2007) 3049-3060

and also the lecture notes by Paul (given at KTH in 2016):
http://www.mcs.anl.gov/~fischer/kth/kth_crs_2016s.pdf

Regarding your questions below: The basis functions are the Lagrange
interpolants to the Legendre polynomials of a specific order. If using
PnPn-2, the velocity is on the Gauss-Lobatto-Legendre mesh (i.e.
including the boundary points), and the pressure is on the
Gauss-Legendre mesh (without boundary points). These functions are
defined within each element, and the continuity between elements is C0,
i.e. only the function value is the same. The ansatz functions are
polynomials, so you can differentiate them inside each element; however,
derivatives are not continuous over element boundaries (even though this
difference reduces spectrally fast).

Hope this helps to get a start on SEM.

Philipp



On 2018-01-04 16:16, nek5000-users at lists.mcs.anl.gov wrote:
> Hi,
>
>
> I am trying to understand how the code works, and therefore I opened the
> theory section at the Nek5000-homepage.
>
> Here, you link to the book "High-Order Methods for Incompressible Fluid
> Flow" by Deville et al.
>
> I started to read the book, but I am not sure how to relate what I read
> to Nek.
>
>
> In chapter 4, a class of 2D basis functions defined on a square is defined.
>
> I think this is the kind of basis function that is used in Nek, am I right?
>
>
> I understand that these functions serves as a basis for one element of
> the domain.
>
> But Nek solves PDE:s over several linked elements.
>
> What is here the condition at the edge connecting two elements?
>
>
> Is it that the function (velocity component/pressure/scalar) should be
> continuous at the GLL-nodes at such an edge?
>
> Also consinuously differentiable att these nodes? Differentiable to some
> higher order?
>
> What about function values on the edge that are not at the GLL-node?
>
> Are they discontinuous or differentiable to some certain order?
>
>
> Best,
>
>
> Johan
>
>
>
>
>
> _______________________________________________
> Nek5000-users mailing list
> Nek5000-users at lists.mcs.anl.gov
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