fiat team mailing list archive

[Anders Logg <logg@xxxxxxxxx>]

You mean the transitive closure of the support? ;-)

--
Anders


On Wed, Sep 16, 2009 at 10:53:41AM -0500, Matthew Knepley wrote:
> I do agree with Anders, that if we associate a single basis function to each
> vertex (P1),
> then that function must have support on the "star" of the vertex, or every cell
> which
> includes that vertex on its boundary.
>
>   Matt
>
> On Wed, Sep 16, 2009 at 10:50 AM, Garth N. Wells <gnw20@xxxxxxxxx> 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
>
>
>
>
>
>
>
>
>

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