[firedrake] Second derivatives

[Justin Chang]

I am looking at equal order mixed formulations. Two discretizations in
particular:

1) Our group recently released a paper on enforcing local mass balance and
discrete maximum principles through the least-squares finite element
method. The paper can found here:

http://arxiv.org/pdf/1506.06099v1.pdf

Equation 4.13 is what I want to solve using firedrake. Everything in that
paper was written in MATLAB because the authors could not get FEniCS to do
what they want. I am guessing it's because there were no Q1 elements
available within FEniCS (double derivative of P1 elements makes no sense).

2) The other discretization, based on the variational multi-scale
formulation, for Darcy-Brinkmann equations is described in this paper:

http://onlinelibrary.wiley.com/doi/10.1002/fld.2544/abstract

Though I am currently looking at the steady-state version of this. Appendix
A1 describes how they discretize the \delta v and \delta w terms.

I could perhaps figure these out through trial and error, but I got my
hands full with other things ATM :)

Thanks,
Justin

On Tue, Jul 21, 2015 at 4:33 AM, David Ham <David.Ham at imperial.ac.uk> wrote:

> Hi Justin,
>
> Can you provide a little more information about the sort of discretisation
> you're talking about: if we can see what you're talking about then we'd be
> in a better position to tell you whether Firedrake can do that.
>
> Cheers,
>
> David
>
> On Fri, 17 Jul 2015 at 07:39 McRae, Andrew <a.mcrae12 at imperial.ac.uk>
> wrote:
>
>>    It's possible to represent second derivatives in a form, such as
>> assemble(div(grad(f))*dx).  I assume this is true in FEniCS as well as
>> Firedrake, because this is just UFL/FFC/FIAT functionality.  This would
>> produce the sum (over cells) of the Laplacian of f on each cell.
>>
>>  However, like FEniCS, none of our function spaces have more than C^0
>> continuity.  That is, the functions are at most continuous, but won't have
>> continuous derivatives.  It's likely that you won't want to use second
>> derivatives of C^0 functions in a practical discretisation (though I'm sure
>> there are methods that *do* do this).
>>
>>  Andrew
>>
>> On 17 July 2015 at 01:50, Justin Chang <jychang48 at gmail.com> wrote:
>>
>>> Hi everyone,
>>>
>>>  Is it possible to do second derivatives? What I mean by that is things
>>> like div[grad[u]] and grad[grad[u]]. I haven't tried this out yet, but we
>>> use these discretizations a lot for our research, and FEniC's inability to
>>> do this made us sad (although our more major qualms had to do with its
>>> inability to support quads).
>>>
>>>  Thanks,
>>> Justin
>>>
>>>
>>
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