[dune-functions] GridFunction interface

[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.

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.

Ciao
Christian