Subject: [Ffc] Bubble function

From: Andy Ray Terrel <[email-masked]>
Date: Fri, 9 Apr 2010 14:46:13 -0500

Is there anything telling how the bubble functions are defined?

For example with the Mini example:

P1 = VectorElement("Lagrange", "triangle", 1)
B = VectorElement("Bubble", "triangle", 3)
Q = FiniteElement("Lagrange", "triangle", 1)
Mini = (P1 + B) * Q

Does B consist of 3 bubble functions in some symmetric pattern in the element?

-- Andy

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From: [email-masked]
Date: Sat, 10 Apr 2010 03:43:49 +0200

> Does B consist of 3 bubble functions in some symmetric pattern in the

The bubble functions are the bubble functions ;)

In the above example, B is the span of the piecewise continuous
cubic functions that are zero on the boundary of each triangle.
(1 per triangle for a FiniteElement and hence 2 per triangle for a
VectorElement of value dimension 2.)

In general, the bubble finite element of degree k is the span of the interior
basis functions of CG_k.

--
Marie

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From: Andy Ray Terrel <[email-masked]>
Date: Sat, 10 Apr 2010 11:14:04 -0500

Okay I see.  Its just a third order bubble, sometimes I send mail
before I think.  The mini element is a first order bubble or at least
that's what Brezzi Fortin say.

-- Andy

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From: Marie Rognes <[email-masked]>
Date: Sun, 11 Apr 2010 13:00:19 +0200

> that's what Brezzi Fortin say.

That depends on your definition of the word "is".

The "velocity" part of the (lowest order) Mini element is, in FFC:
   
    P1 = VectorElement("Lagrange", triangle, 1)
    B = VectorElement("Bubble", triangle, 3)
    V = P1 + B

The logic here is that the degree should be such that any element
of degree k is included in P_k. Hence, for triangles, the numbering for the
bubbles start at 3.

--
Marie

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From: Andy Ray Terrel <[email-masked]>
Date: Sun, 11 Apr 2010 18:26:56 -0500

Okay so Brezzi Fortin have just a bubble function with one node.
Perhaps they meant P3 but their notation is ambiguous.
ArnoldBrezziFortin1984 does call for the P3 in the original paper but
then they mention using 2 P4 bubble functions for the next velocity
order.  If I do enrich with 2 P4 bubbles with ffc will the bubble
functions be any different.  It just seems like it will give exactly
the same evaluation points as enriching with 1 bubble function.

-- Andy

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From: Marie Rognes <[email-masked]>
Date: Mon, 12 Apr 2010 08:51:30 +0200

> the same evaluation points as enriching with 1 bubble function.

I'm not quite sure what you mean by enriching with 2 P4 bubble 
functions. Do you mean 2 as in 2 components? Note that the bubble space 
of degree 4 has local dimension 3.

Are you sure  that you have the latest ffc?

--
Marie

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From: Andy Ray Terrel <[email-masked]>
Date: Mon, 12 Apr 2010 09:28:37 -0500

On Mon, Apr 12, 2010 at 1:51 AM, Marie Rognes <[email-masked]> wrote:
> I'm not quite sure what you mean by enriching with 2 P4 bubble functions. Do
> you mean 2 as in 2 components? Note that the bubble space of degree 4 has
> local dimension 3.

Hmm perhaps they just mean 2 components, funny they stress 2 function
for P4 but not for P3.  What do you mean by local dimension 3?  Do you
mean the spatial dimension or 3 basis functions?

> Are you sure  that you have the latest ffc?

Yeah.  But I can put anything 3 and above as the order and it looks to
work.  I just want to make sure I have some vague notion of what I am
doing.