ffc team mailing list archive

[Anders Logg <logg@xxxxxxxxx>]

On Wed, Sep 16, 2009 at 05:37:13PM +0200, Garth N. Wells wrote:
>
>
> Anders Logg wrote:
> > On Wed, Sep 16, 2009 at 04:34:29PM +0200, Garth N. Wells wrote:
> >>
> >> Anders Logg wrote:
> >>> @Garth: Yes, they are defined everywhere. For example, v1 = 1 - x - y,
> >>> v2 = x and v3 = y are a nodal basis for the reference triangle, but
> >>> they are also a basis for P1 on R^2. You can evaluate 1 - x - y for
> >>> any values of x and y.
> >>>
> >> I don't agree at all. The function 1-x-y can be evaluated anywhere, but
> >> the basis functions are defined only on the cell. If the basis functions
> >>   were defined everywhere, the method would loose all sparsity.
> >
> > The sparsity is not a result of evaluate_basis returning 0 outside the
> > cells, it's a result of the assembly process and the local-to-global
> > mapping.
> >
> > Either way, that's not how evaluate_basis is implemented today. It
> > works perfectly fine to evaluate the natural extensions of the basis
> > functions, except along one line where the mapping happens to be
> > singular. This is not in anyway connected to how the basis functions
> > should be defined, it's just a consequence of the particular way Rob
> > has implemented the basis functions in FIAT (via a mapping from Jacobi
> > polynonials on a square).
> >
> > For example, if you take the first basis function (1 - x - y) on the
> > reference triangle and evaluate it at (2, 2), you get -2 as
> > expected although that point is outside the triangle. But if you try
> > to evaluate it at (1, 1), you get 0 because of a bug/feature in
> > evaluate_basis. If you do (1, 1 + eps), then you get -1 - eps.
> >
>
> The sparsity is a result of the basis being non-zero only locally, so I
> would call getting zero outside of the cell from ufc::evaluate_basis(..)
> a feature.

That's true only for DG elements. All other bases we know of have
larger support (on a patch of elements).

Furthermore, evaluate_basis does currently *not* return zero outside
the cell, only for *certain* points outside the cell.

--
Anders

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