[Nek5000-users] Basis functions and smoothness of interpolants

Mon Jan 8 08:16:41 CST 2018

Hi,
The MATLAB codes used to be on the homepage of the Linné FLOW Centre, 
but due to some re-structuring the old pages are currently not available.

Regarding time integration:
- linear terms: implicit backward differencing schemes (BDF) of up to 
order 3
- nonlinear terms: explicit extraploation scheme (EXT) up to order 3.

Best regards,
Philipp

On 2018-01-08 10:09, nek5000-users at lists.mcs.anl.gov wrote:
> 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
>> 
>> 
>> 
>> 
>> 
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>> Nek5000-users at lists.mcs.anl.gov
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>> 
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