fiat team mailing list archive

[Jed Brown <jed@xxxxxxxx>]

I was playing with FIAT today and became somewhat confused with the choice of
Lagrange basis.  I'm just working in one dimension because I'm building high
order tensor products bases.

It appears that the Lagrange basis is based on evenly spaced points, but that's
terrible if the polynomial order grows.  Indeed, when I plotted the basis for
modest order, the Runge effect quickly becomes a problem.  So why is the
Lagrange basis not imeplemented on Legendre-Gauss-Lobatto nodes?  It's not
really harder to do since

  jacobi.compute_gauss_jacobi_points(1,1,p-1)

gives the interior points of an order p basis).  They are pretty much the same
at order 3, but by order 4 there is already an advantage to using the LGL
points.  Is there any advantage (ever) to using equally spaced nodes?  If you
want the points to nest, the Chebyshev points would be fine.  Am I missing
something?  I think it would be nice to be able to choose LGL/CGL/equal spacing,
but this requires some (maybe not much) refactoring.  Is it sufficient to have
make_lattice_xxx() take an extra parameter and add corresponding plumbing up
to the front-ends?

On another note, in K&S, the `standard modal' basis is chosen so that the
element mass and stiffness (for linear Laplacian) matrices are banded.  This
might be a nice basis to use for p-type refinement, but unfortunately it's
sparseness doesn't hold for nonlinear problems.  Does anyone (potential FIAT
users) care about hierarchical bases and sparser matrices for high order
Laplacians?

Jed

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