fenics team mailing list archive

[Anders Logg <logg@xxxxxxxxx>]

On Thu, Apr 29, 2010 at 02:53:16PM +0100, Garth N. Wells wrote:
>
>
> On 29/04/10 14:41, Anders Logg wrote:
> >On Thu, Apr 29, 2010 at 02:19:44PM +0100, Garth N. Wells wrote:
> >>
> >>
> >>On 29/04/10 11:03, Anders Logg wrote:
> >>>As you are all aware, FEniCS uses a somewhat unorthodox convention for
> >>>the order of test and trial function, which is to place the test
> >>>function first:
> >>>
> >>>   a(v, u)
> >>>
> >>>At some point in history, u was chosen as a name for the unknown and v
> >>>was chosen as a name for the test function. At another point in
> >>>history, i was chosen as the name of the row index in a matrix and j
> >>>was chosen as a column index. So people tend to write
> >>>
> >>>   a(u, v)
> >>>   A_ij
> >>>
> >>>This is somewhat unfortunate, since i (rows, equations) corresponds to
> >>>the test function v and j corresponds to the trial function u.
> >>>
> >>>So we must define the matrix A by
> >>>
> >>>   A_ij = a(phi_j, phi_i)
> >>>
> >>>Notice the swapping of i and j here. This might not be much of a
> >>>problem but it's certainly a problem when we consider multilinear
> >>>forms and their corresponding tensors (as we do frequently in FEniCS):
> >>>
> >>>   A_i = a(..., phi_i3, phi_i2, phi_i1)
> >>>
> >>>For this reason, Rob and I decided to change the order of u and v and
> >>>write
> >>>
> >>>   a(v, u)
> >>>
> >>>when we wrote the first FFC paper back in 2005. Rob is not so much to
> >>>blame for this, but I've been pushing this quite hard since then with
> >>>the result that v is before u in the current draft version of the
> >>>FEniCS book, as well as in all demos.
> >>>
> >>>The question is now, should we stick by this? The reason for doing so
> >>>are:
> >>>
> >>>   1. It the logic thing: v<-->   i and u<-->   j
> >>>
> >>>   2. UFL, UFC and the DOLFIN assembler are all based on this
> >>>
> >>>The reasons for not doing so are:
> >>>
> >>>   1. a(u, v) looks better (alphabetical ordering)
> >>>
> >>>   2. It's what most people are used to
> >>>
> >>>If we choose this second option, we need to define the order of
> >>>arguments in a form to start from the right in all text that relates
> >>>to finite element assembly (in the FEniCS book) and in all code that
> >>>deals with forms (UFL, UFC, DOLFIN). Note that this means we don't
> >>>need to change any code at all, but the text would need changing.
> >>>In particular, the UFC function create_finite_element(0) would still
> >>>return the finite element for the test space so that the assembler
> >>>would assemble A and not A transposed.
> >>>
> >>>So for a bilinear form
> >>>
> >>>   a(u, v)
> >>>
> >>>we would need to say that v is the first argument (numbered 0) and u
> >>>would be the second argument (numbered 1).
> >>>
> >>>And for a semilinear form (see FEniCS book chapter on forms), we would
> >>>need to write
> >>>
> >>>   F(v_r, ..., v_3, v_2, v_1; w_1, w_2, w_3, ..., w_n)
> >>>
> >>>where {v_i} are arguments (basis functions) and {w_i} are
> >>>coefficients.
> >>>
> >>>This is a bit awkward (and the reason for changing the order) in the
> >>>first place, but this would only appear deep down in the more
> >>>technical parts of the text and so most users would not see it. They
> >>>will only see the swapping of i and j when defining the matrix for
> >>>Poisson.
> >>>
> >>>I hope we can have a long and good discussion on this so we settle
> >>>this once and for all. All comments/votes are welcome.
> >>>
> >>
> >>I don't feel particularly strongly either way, but have a slight
> >>leaning to go with a(u, v). Both a(u, v) and a(v, u) are commonly
> >>used, and which is more common depends on the field.
> >
> >Where is a(v, u) used? I only hear comments that a(u, v) is used
> >*everywhere* else and we need to change.
> >
>
>
> Here are two 'a(v, u)' books from a quick scan of my book shelves
>
>   - Hughes, TJR, The Finite Element Method
>   - Donea and Huerta, Finite Element Methods for Flow Problems
>
> Engineering numerical analysis and mechanics book often use a(v, u),
> I assume because when considering the weight function as a
> 'variation', it's common to use the order
>
>    \delta u u
>
> Garth

ok, so they did the right thing... :-)

Another thought is this: u should come before v since u is more
important. Same goes for matrices, columns before rows since a matrix
is really a list of column vectors (the range of the matrix is
spanned by its columns), so we should really be changing to A_ji to
match a(u, v)...

--
Anders

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