[dune-functions] GridFunction interface

[Carsten Gräser]

Hi Christian,

Am 21.03.2015 um 20:42 schrieb Christian Engwer:
> Hi Carsten,
> 
>> For me the local function is given by g = f \circ T.
>> Then range(g) = range(f) regardless if f is an analytic function,
>> or Pk or RT0 FE function in flat or curved space.
>> 1) and 2) is only about the derivative. For 2) its just g'
>> whereas its f' \circ T != g' for 1). But this can still be
>> interpreted as g' in a special coordinate system.
>>
>> Notice that, with this definition the local function will be a linear
>> combination of the local FE, for some FE spaces (like Pk) but not for
>> others (like RT).
>>
>> What's your precise definition of a local function?
>> Since you're taking about a change in the range it must
>> be different from both, 1) and 2).
> 
> OK, in this sense we agree. That's exactly the reason why I asked Oli
> about whether the local function would apply the piola
> transformation. If the transformation is not applied, like Oli
> suggests, we would always have the local function being a linear
> combination of the local finite shape functions. In this case (that
> was what I implicitly assumed) the ranges have to differ in some
> cases.
maybe this was a miss-understanding. As far as I know the Piola
transformation incorporates the transformation of range _and_ domain.
In this sense it's not applied because the domain is not transformed.
But now it's clear what we're taking about.

> I would be fine with your definition. Although I would like to think
> in more detail about possible performance consideration of the
> different approaches.
> 
> In addition to how should a function behave, we should also discuss
> the context of local functions. I think the local functions should
> behave the same way as local shape functions in the assembler. I think
> having everything in the local coordinate system, but not as a linear
> combination of shape functions is feasible, but I would still like to
> think about it further.
I guess you mean the 'global finite elements'. In approach 1) the local
function will behave like this.

Carsten