ffc team mailing list archive

[Anders Logg <logg@xxxxxxxxx>]

On Wed, Sep 16, 2009 at 07:58:26PM +0200, Garth N. Wells wrote:
>
>
> Anders Logg wrote:
> > On Wed, Sep 16, 2009 at 12:57:52PM -0400, Shawn Walker wrote:
> >> On Wed, 16 Sep 2009, Anders Logg wrote:
> >>
> >>> 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.
> >>>
> >> This discussion doesn't concern me, but here is my opinion.
> >>
> >> I think that it should be possible for a user to evaluate the basis
> >> functions outside the reference element.  This could potentially be
> >> useful for a generalized FEM.  Or perhaps a user wants to `hack'
> >> DOLFIN to do some peculiar thing, for whatever reason.  This should
> >> be allowed!
> >>
> >> I think in general, one should not limit functionality unless there
> >> is a strong reason otherwise.  You could still output a warning
> >> message.
> >
> > A share this view.
> >
>
> What is the isoparametric map outside of the cell?

The map is a polynomial and so are the basis functions so they
naturally extend to any point in the universe.

--
Anders

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