ffc team mailing list archive

[Anders Logg <logg@xxxxxxxxx>]

On Wed, Sep 16, 2009 at 06:12:48PM +0200, Garth N. Wells wrote:
>
>
> Anders Logg wrote:
> > On Wed, Sep 16, 2009 at 06:02:51PM +0200, Garth N. Wells wrote:
> >>
> >> Anders Logg wrote:
> >>> On Wed, Sep 16, 2009 at 05:50:05PM +0200, Garth N. Wells wrote:
> >>>> Anders Logg wrote:
> >>>>> 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).
> >>>>>
> >>>> We can define the basis on a cell to be zero outside of the cell. The
> >>>> dof map takes care of patching together the functions on neighbouring cell.
> >>>>
> >>>>> Furthermore, evaluate_basis does currently *not* return zero outside
> >>>>> the cell, only for *certain* points outside the cell.
> >>>>>
> >>>> It would be nice if it returned zero outside the cell, but it is a
> >>>> relatively low-level function so there may be efficiency reasons why
> >>>> this isn't desirable, in which case it wouldn't both me if the returned
> >>>> values for points outside the cell are meaningless.
> >>>>
> >>>> Garth
> >>> Yes, that could be an option, or adding to the manual that the values
> >>> are undefined outside the cell.
> >>>
> >>> It's just that it would be very useful for us (Marie and I) to reuse
> >>> the cell basis functions on cell patches.
> >> Are you wanting to evaluate functions like v2 = x outside of [0, 1]?
> >> What do you do with it?
> >
> > I want to have a basis for say P2 on a patch of elements surrounding an
> > element K and one option would be to use the extension of the P2
> > basis on K.
> >
>
> I don't understand how evaluating the P2 basis for K outside of K would
> work. For example, it could be greater than one.

Yes, it can be greater than one but that's not a problem. The point is
that the extension of the basis for P2 on K spans P2 on R^2.

Just think of the nodal basis for P1 on the reference triangle:

  v1 = 1 - x - y
  v2 = x
  v3 = y

These three functions span P1 on R^2 or on any other triangle than K.
But they are only a *nodal* basis on K.

--
Anders

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