On 02/06/2012 07:45 PM, John Peterson wrote:
> On Mon, Feb 6, 2012 at 4:36 PM, David Knezevic
> <dknezevic@...> wrote:
>> On 02/06/2012 06:09 PM, John Peterson wrote:
>>> On Mon, Feb 6, 2012 at 3:24 PM, David Knezevic
>>> <dknezevic@...> wrote:
>>>> Hi all,
>>>> I'd like to use a DG version of the Lagrange shape functions. This is
>>>> pursued, for example, in the book "Nodal Discontinuous Galerkin Methods"
>>>> by Hesthaven and Warburton.
>>> What's the main motivation for using the Lagrange basis? Better
>>> conditioning than monomials?
>> I'd like to compare to a Matlab DG code which uses (or will use) these
>> L2_LAGRANGE basis functions. These basis functions are important in Matlab
>> because with, say, MONOMIALs, you need to do an L^2 projection to represent
>> f(u_h) in the FE space. The element loop to assemble the right-hand side for
>> this projection appears to be a bottleneck in Matlab, especially if you have
>> to do it every timestep. With L2_LAGRANGE you can do interpolation instead
>> of projection.
> Oh, that sounds like a good idea actually...
>> Also, yep, conditioning of L2_LAGRANGE will be better than MONOMIALs for the
>> same order shape functions. But (L2_)LAGRANGE only goes up to cubic and the
>> condition number for cubic MONOMIALs isn't too bad.
> We could probably add higher orders too if this is determined to be useful...
I guess the one thing to be careful about is equi-spaced points aren't
very good for high order (Runge's phenomenon et al.), but as usual
they'd be fine for reasonably low order.
The book I mentioned earlier discusses how to construct Chebyshev-like
nodal locations on simplices, and they go up to 24^th order polynomials
> I assume you can directly call the continuous LAGRANGE shape() and
> shape_deriv() functions for the discontinuous family?
Yep, that should be identical. I guess the only change is in the number
of global dofs and the dof numbering, which would be taken care of by
Roy's suggestion of associating dofs to elements rather than nodes.