[petsc-users] mathematics formula of DMPlexProjectFunction

[Matthew Knepley]

On Tue, Oct 20, 2015 at 1:32 PM, Fande Kong <fdkong.jd at gmail.com> wrote:

> Jed, Thanks.
>
> It means that the inner product against basis "n_i" is just the function
> value at the point "x_i" if the function is sufficiently regular, e.g.,
> sin(x) or cos(x).
>

No, n_i is not part of the basis for the approximation space, its a basis
vector for the dual space. Thus its
not an inner product, it the dual pairing. Here we can always use an
integral since the dual elements can
be represented by measures

  Riesz-Markov-Kakutani Representation Theorem (
https://en.wikipedia.org/wiki/Riesz%E2%80%93Markov%E2%80%93Kakutani_representation_theorem
)

    Matt


> The basis function at x_i is "x-x_i" which is not the one we use
> to discretize the equations?
>
>
> On Tue, Oct 20, 2015 at 12:03 PM, Jed Brown <jed at jedbrown.org> wrote:
>
>> Fande Kong <fdkong.jd at gmail.com> writes:
>> > Any body knows the mathematics formula corresponding to the function
>> > DMPlexProjectFunction? I already went through the code, but I do not
>> > understand quite well. I will appreciate any help.
>>
>> The definition of a finite element involves a dual space (the basis for
>> which is sometimes called the "nodes").  For a typical finite element
>> space, the "nodes" are Dirac delta functions at vertices.  Consequently,
>> the inner product
>>
>>   (n_i, f) = \int \delta(x - x_i) f(x) = f(x_i)
>>
>> if f is sufficiently regular.  For more general dual basis functions,
>> this inner product needs to be evaluated by quadrature.
>> DMPlexProjectFunction does this projection.
>>
>
>


-- 
What most experimenters take for granted before they begin their
experiments is infinitely more interesting than any results to which their
experiments lead.
-- Norbert Wiener
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